universidade do minhorepositorium.sdum.uminho.pt/bitstream/1822/48656/1/daniel antoni… · vii...
TRANSCRIPT
Universidade do MinhoEscola de Ciências
Daniel António da Silva Miranda
fevereiro de 2017
Optimizing performance of rechargeable lithium-ion batteries through computer simulations
Dan
iel A
ntón
io d
a Si
lva
Mira
nda
Op
tim
izin
g p
erf
orm
an
ce o
f re
cha
rge
ab
le li
thiu
m-io
n
ba
tte
rie
s th
rou
gh
co
mp
ute
r si
mu
lati
on
s U
Min
ho|2
017
Universidade do MinhoEscola de Ciências
Daniel António da Silva Miranda
fevereiro de 2017
Optimizing performance of rechargeable lithium-ion batteries through computer simulations
Trabalho efetuado sob a orientação doProfessor Doutor António Mário Lourenço da Fonseca Almeida
doProfessor Doutor Senentxu Lanceros-Méndez
e daProfessora Doutora Maria Manuela da Silva Pires da Siva
Tese de Doutoramento em CiênciasEspecialidade em Física
v
To my parents for everything
To my wife and my daughter for existing
“Não existem sonhos impossíveis para aqueles que realmente acreditam que o
poder realizador reside no interior de cada ser humano. Sempre que alguém descobre
esse poder, algo antes considerado impossível, se torna realidade.”
Albert Einstein
vii
Acknowledgements
To my supervisor Professor Mário Almeida, I appreciate the opportunity to work with
him, all transmitted scientific knowledge and friendship built over the years of working
together.
To my co-supervisors Professor Senentxu Lancers-Méndez and Maria Manuela Silva,
thank you for all the support and helping me whenever I needed.
To my friend Carlos Costa for his availability and support.
To all the colleagues of the ESM group who helped me and encouraged. I have only
these words: thank you very much.
To my family and friends who directly or indirectly contributed to this work and are not
mentioned.
To my Parents, António and Rosa, my brother Francisco, my wife Vânia and my
daughter Maria, my parents in law Ilda and Artur for all that signify to me: my strength
to live.
ix
Abstract
There is an increasing need for larger battery autonomy and performance related to
rapid technological advances in portable electronic products such as mobile-phones,
computers, e-labels, e-packaging and disposable medical testers, among others.
The advantages of lithium-ion batteries in comparison to other battery types, such
as Ni-Cd ones, are the fact of being lighter and cheaper, showing high energy density
(between 100 and 150 Wh kg-1) and a large number of charge/discharge cycles.
The key issues for improving lithium-ion battery performance are specific energy,
power, safety and reliability. Typically, the performance of a battery is optimized for
either power or energy density through the improvement of electrodes and separator
materials.
Computer simulations of battery performance are important and critical for
optimizing materials and geometries. Models have been developed considering the
physical-chemical properties of the materials to be used as electrodes and separators, the
choice of the most suitable organic solvents for electrolytes, the geometry and
dimensions of the components that make up the battery as well as the porosity of the
electrodes.
The objective of the present work was the optimization of lithium-ion battery
performance through computer simulations based on the Doyle/Fuller/Newman model
for separators, electrodes (anode and cathode) and full/half-cells in order to understand
the main processes that affect battery performance.
Thus, along this work, simulations were developed to improve the performance of a
lithium-ion batteries. Thus, simulation of the different battery components (separator
and electrodes) were developed. The first simulation explores the influence of the
geometrical parameters of the separator (porosity, turtuosity and separator thickness) in
the performance of the battery. Then, the optimal relationship between active material,
binder and conductive additive for lithium-ion battery cathode was studied. Further, a
simulation of an interdigitated battery was performed, where the effect of the number,
thickness and the length of the digits on the delivered battery capacity was evaluated.
Finally, different conventional and unconventional geometries were evaluated taking
into account their suitability for different applications without and with consideration of
x
different thermal conditions. The different thermal conditions included isothermal,
adiabatic, cold, regular and hot conditions.
In relation to the separator, it was observed that its ionic conductivity depends on
the value of the Bruggeman coefficient, which is related to the degree of porosity and
tortuosity of the membrane. It was determined that the optimal value of the degree of
porosity is above 50% and the separator thickness should range between 1 μm and 32
μm for improved battery performance.
For the electrodes, it is shown that optimization of the electrode formulation is
independent of the active material type but depends on the minimum value of n, defined
as the percentage of binder content /percentage of conductive material, depending also
on the discharge rate.
The influence of different geometries, including conventional, interdigitated,
horseshoe, spiral, ring, antenna and gear, in the performance of lithium-ion batteries was
analyzed and the delivered capacity depends on geometrical parameters such as the
maximum distance that ions move until occurs intercalation process, the distance
between the current collectors and the thickness of the separator and the electrodes.
For interdigitated structures, the delivered capacity of the battery increases with
increasing the number of digits as well as with increasing thickness and length of the
digits.
Finally, the influence of the thermal behavior on battery performance was evaluated
for the aforementioned geometries under different conditions, isothermal, adiabatic,
cold, regular and hot conditions. The gear and interdigitated batteries presented the
highest delivery capacity at all thermal conditions.
In conclusion, in order to improve the performance of lithium ion batteries, it is
necessary optimize the geometric parameters of the separator, the percentages of binder,
active material and conductive additive in the cathode, as well as the battery geometry
(conventional, interdigitated and unconventional geometries) at different thermal
conditions.
xi
Resumo
Nos dias de hoje, devido ao galopante avanço tecnológico, há uma crescente
necessidade de maior autonomia e desempenho de baterias para uso em dispositivos
eletrónicos portáteis (telemóveis, computadores, identificadores eletrónicos, embalagens
eletrónicas e dispositivos médicos de diagnóstico descartáveis, etc).
As vantagens das baterias de iões de lítio, comparativamente com outros tipos de
baterias tais como as de Ni-Cd, são o facto de serem mais leves e económicas, tendo
elevada densidade de energia (entre 100 e 150 Wh kg-1) e um elevado número de ciclos
de carga/descarga.
As caraterísticas que permitem potenciar o desempenho da bateria de iões de lítio
são energia específica, potência, segurança e confiabilidade. Tipicamente o desempenho
de uma bateria é otimizado para uma melhor potência ou densidade de energia, o que é
conseguido através da melhoria dos elétrodos e do material dos separadores.
As simulações computacionais que avaliam o desempenho das baterias são de uma
enorme importância para a otimização de materiais e geometrias das mesmas. Os
modelos têm sido desenvolvidos tendo em conta as propriedades físico-químicas dos
materiais que são usados como elétrodos e separadores, a escolha de solventes
orgânicos mais adequados para os eletrólitos, a geometria e dimensões dos componentes
que constituem a bateria, assim como a porosidade dos elétrodos.
O objetivo do meu trabalho é a otimização do desempenho da bateria de iões de
lítio através de simulações em computador baseadas no modelo de
Doyle/Fuller/Newman para os separadores, elétrodos (ânodo e cátodo) e
completas/meias-células de baterias de iões de lítio, para que se possam entender os
principais processos que afetam o desempenho da bateria.
Assim, ao longo deste trabalho, foram desenvolvidas simulações para melhorar o
desempenho das baterias de iões de lítio, tendo sido implementadas simulações dos
diferentes componentes da bateria (separador e elétrodos). Numa primeira simulação
explorou-se a influência dos parâmetros geométricos do separador (porosidade,
tortuosidade e espessura do separador) no desempenho da bateria. De seguida, fez-se
um estudo otimizado da relação entre o material ativo, o material ligante e o material
condutor para o cátodo de uma bateria de iões de lítio. Além disso, foi realizada uma
simulação de uma bateria interdigitada, onde foi avaliado o efeito do número, espessura
e comprimento dos dígitos na capacidade da bateria. Finalmente foram avaliadas
xii
diferentes geometrias convencionais e não convencionais tendo em conta a sua
adequação para diferentes aplicações, considerando diferentes condições térmicas. As
diferentes condições térmicas incluíram condições isotérmicas, adiabáticas, frias,
regulares e quentes.
Em relação ao separador observou-se que a condutividade iónica depende do valor
do coeficiente de Bruggeman, que está relacionado com o grau de porosidade e
tortuosidade da membrana. Assim, foi determinado que o melhor valor para o grau de
porosidade se situa acima de 50% e que a espessura do separador se deve situar entre 1
μm e 32 μm, para um melhor desempenho da bateria.
Para os elétrodos mostrou-se que a sua otimização é independente do tipo de
material ativo, mas depende do valor mínimo de n, razão entre a percentagem de
material ligante (C2) e material condutor (C3), dependendo também da taxa de descarga.
A influência das diferentes geometrias (convencional, interdigitada, ferradura,
espiral, anel e roda dentada) no desempenho das baterias de iões de lítio foi analisada e
o seu valor de capacidade depende de parâmetros geométricos tais como, a distância
máxima que os iões se movem até que ocorra processo de intercalação, distância entre
os coletores de corrente e a espessura do separador e elétrodos.
Para a geometria interdigitada a capacidade da bateria aumenta, não só com o
aumento do número de dígitos, mas também com o aumento da espessura e do
comprimento dos dígitos.
Por fim, a influência do comportamento térmico no desempenho da bateria sob
diferentes condições (condição isotérmica, adiabática, frio, regular e quente) foi também
avaliada para as diferentes geometrias. Neste aspeto, as geometrias roda dentada e
interdigitada foram as que revelaram maior valor de capacidade para todas as condições
térmicas.
Em conclusão, no sentido de aumentar o desempenho das baterias de iões de lítio é
necessário otimizar os parâmetros geométricos do separador, as percentagens de
material ligante, material ativo e material condutor no cátodo, bem como a geometria da
bateria (convencional, interdigitada, e as não convencionais), para diferentes condições
térmicas.
xiii
List of Symbols and Abbreviations
a specific interfacial area, m2/m3
Aaa area of active material in the not interdigitated part of the anode (m2)
Abat cell cross section area, m2
Acc1 area of active material in the not interdigitated part of the cathode (m2)
Ai area of a given component in the battery i(i = a,s,c)
brugg Brugg parameter in the electrodes
c_dig digit length of the electrode, m
CE concentration of Li ions in the electrode, mol/m3
CL concentration of Li ions in the electrolyte, mol/m3
Cp,i heat capacity at constant pressure of battery components i( i = a,s,c), J/(kg.K)
CNT carbon nanotubes
C1 percentage of the active material, %.
C2 percentage of the binder, %.
C3 percentage of carbon black, %.
D diffusion coefficient of the salt in the electrolyte, m2/s
d_cc distance between of collectors
DEC diethyl carbonate
DLI diffusion coefficient of Li ions in the electrode, m2/s
d_max maximum distance of ions from the collector positions
DMC dimethyl carbonate
DMPU N,N’-dimethyl propylene urea
D1 density of the active material, g/m3.
D2 density of the binder, g/m3.
D3 density of carbon black, g/m3.
Ead,i activation energy for diffusion in the electrodes i(i = a,c), J/mol
Eak,i activation energy for reaction in the electrodes i(i = a,c), J/mol
EC ethylene carbonate
e_dig digit thickness of the electrode, m
EMC ethyl methyl carbonate
EV electric vehicles
e_sep separator thickness, m
F Faraday’s constant, 96487 C/mol
FEA finite element analysis
FEM finite element method
xiv
f activity of the salt in the electrolyte, mol/m3
grade GF/A
Whatman glass microfiber filters
h heat transfer coefficient, W/(m2.K)
HEV hybrid electric vehicles
HOPG highly ordered pyrolytic graphite
iE current density in the electrode, A/m2
iL current density in the electrolyte, A/m2
ITOTAL total current density, A/m2
jLi+ pore wall flux of Li ions, mol/cm2 s
L width, m
L_dim dimension of horseshoe, m
LFP lithium iron phosphate
LMO lithium manganese oxide
LTO lithium titanium oxide
M mass transport flux, mol/m2
MCMB mesocarbon microbeads
mTotalc total mass of cathode, g
N number of digits for interdigitated and gear battery
NMP N-methyl-2-pyrrolidone
ODE ordinary differential equation
p porosity of the separator
PAN poly(acrylonitrile)
PC propylene carbonate
PCM cooling phase change materials cooling
PDA principal differential analysis
PE poly(ethylene)
PEO poly(ethylene oxide)
PVDF poly (vinylidene fluoride)
PVDF-TrFE poly(vinylidene-co-trifluoroethylene)
PVDF-HFP poly(vinylidene fluoride-co-hexafluoropropene)
PVDF-CTFE poly(vinylidenefluoride-co-chlorotrifluoroethylene)
Qohmic,i total ohmic heat generation rate of battery components i ( i = a,s,c), W/m3
Qreaction,i total reaction heat generation rate of electrodes i ( i = a,c), W/m3
Qreversible total reversible heat generation rate of electrodes i ( i = a,c), W/m3
Qtotal,i total heat generation rate of battery components i ( i = a,s,c), W/m3
R reaction term of the mass balance equation, mol/m3 s
xv
R gas constant, 8,314 J/mol K
r radius of the electrode spherical particles, m
Rd radius of ring geometry, m
Rf film resistance, m2
Rg radius of gear geometry, m
Super P-C45 carbon black
T temperature, K
t time, s
TMS thermal management system
T external temperature, K
0
t transport number of the positive ions
0u open circuit voltage, V
VTotalc total volume of cathode, m3
W weight of the sample per unit area, g/m2
1D one dimension
2D two dimension
3D three dimension
Greek symbols
i porosity of the region i (i = a,s,c)
f,i volume fraction of the fillers in the electrode i ( i = a,s,c)
over-potential, V
i thermal conductivity of battery components i (i = a,s,c), W/(m.K)
l ionic conductivity of the electrolyte, S/m
ef,i effective ionic conductivity of the electrolyte i (i = a,c), S/m
f effective ionic conductivity of the separator polymer film, S/m
i density of battery components i (i = a,s,c), kg/m3
electronic conductivity of the solid phase of the electrode i (i = a,s,c), S/m
ef,i effective electronic conductivity of the solid phase of the electrode i (i =
a,s,c), S/m
3Pure electronic conductivity of the carbon black (conductive material), S/m
tortuosity of the separator
E potential of the electrodes, V
L potential of the electrolyte, V
xvi
i volume fraction of the material i (i=1, 2, 3)
Subscripts referring specific components of the battery and initial conditions
a anode
adi adiabatic condition
c cathode
cc current collector
cold cold condition
hot hot condition
reg regular condition
s separator
0 initial condition
1 active material
2 binder
3 carbon black
xvii
Table of contents
List of figures ............................................................................................................ xxi
List of tables ........................................................................................................... xxvii
1. Introduction ................................................................................................................ 1
1.1 Introduction ............................................................................................................ 3
1.1.2 Advantages and disadvantages of lithium ion batteries ...................................... 5
1.1.3 Mathematical model for lithium ion batteries ..................................................... 7
1.1.4 Materials ............................................................................................................. 9
1.1.4.1 Anode and cathode electrodes ......................................................................... 9
1.1.4.2 Battery separator ............................................................................................ 10
1.1.5 General mathematical framework for the microscopic models of lithium-ion
batteries ...................................................................................................................... 11
1.2 Objectives ............................................................................................................ 17
1.3 Thesis structure and methodology ....................................................................... 18
1.4 References ............................................................................................................ 20
2. State of the art on microscopic theoretical models and simulations of lithium-ion
rechargeable batteries .................................................................................................... 29
2.1 Microscopic modelling of lithium ion batteries ................................................... 31
2.2 Simulation of the components of the battery: electrodes and separator/electrolyte
................................................................................................................................... 33
2.2.1 Electrodes .......................................................................................................... 33
2.2.2 Separator and electrolyte .................................................................................. 43
2.3 Thermal behavior simulation ............................................................................... 45
2.4 Conclusions .......................................................................................................... 48
2.5 References ............................................................................................................ 50
3. Simulation of Lithium-ion Batteries: Methodology and Theoretical Models ........ 55
3.1 Simulation of lithium-ion batteries ...................................................................... 57
3.1.1 Methodology ..................................................................................................... 57
3.1.2 Development and execution of the simulation ................................................. 58
xviii
3.2 Theoretical models of lithium-ion batteries: Electrochemical and Thermal models
................................................................................................................................... 62
3.3 References ............................................................................................................ 65
4. Modelling separator membranes physical characteristics for optimized lithium ion
battery performance ....................................................................................................... 67
4.1 Introduction .......................................................................................................... 69
4.2 Theoretical model ................................................................................................ 71
4.2.1 General model ................................................................................................... 71
4.2.2 Separator ........................................................................................................... 73
4.3 Parameters and simulation model ........................................................................ 74
4.4 Results and Discussion ........................................................................................ 76
4.4.1 Effect of separator/electrolyte ........................................................................... 76
4.4.2 Effect of the variation of separator membrane physical parameters on battery
performance ............................................................................................................... 78
4.4.2.1 Degree of porosity ......................................................................................... 78
4.4.2.2 Tortuosity ....................................................................................................... 80
4.4.2.3 Dimension/thickness ...................................................................................... 82
4.5 Conclusions .......................................................................................................... 84
4.6 References ............................................................................................................ 85
5. Theoretical simulation of the optimal relationship between active material, binder
and conductive additive for lithium-ion battery cathodes ............................................ 89
5.1 Introduction .......................................................................................................... 91
5.2 Preparation and characterization of the cathodes ................................................ 92
5.3 Theoretical simulation model and model parameters .......................................... 93
5.4 Results and discussion ......................................................................................... 96
5.4.1 LFP and LMO half-cells: validation of the theoretical model .......................... 96
5.4.2 Influence of the cathode components content in the performance of the half-
cell. ............................................................................................................................. 98
5.4.3 Impedance of the LFP and LMO half-cells ..................................................... 103
5.4.4 Electrolyte and Electrode Current Density for LFP half-cells ........................ 106
5.5 Conclusions ........................................................................................................ 111
5.6 References .......................................................................................................... 112
xix
6. Computer simulation evaluation of the geometrical parameters affecting the
performance of two dimensional interdigitated batteries ........................................... 115
6.1 Introduction ........................................................................................................ 117
6.2 Theoretical simulation model and parameters ................................................... 119
6.3 Results ................................................................................................................ 124
6.3.1 Conventional geometry ................................................................................... 124
6.3.2 Interdigitated geometry ................................................................................... 127
6.3.2.1 Influence of the number of digits at different scan rates ............................. 127
6.3.2.2 Influence of length and thickness of the digit .............................................. 130
6.3.2.2.1 Influence of digit length from 60 μm to 480 μm ...................................... 130
6.3.2.2.2 Influence of the digit thickness from 10 μm to 70 μm ............................. 132
6.3.2.2.3 Maximum limit for digit thickness and length at 200C and 400C............ 133
6.4. Discussion ......................................................................................................... 136
6.5. Conclusions ....................................................................................................... 137
6.6 References .......................................................................................................... 138
7. Computer simulations of the influence of geometry in the performance of
conventional and unconventional lithium-ion batteries ............................................ 141
7.1 Introduction ........................................................................................................ 143
7.2 Theoretical simulation model and specific parameters for each geometry ....... 145
7.3 Results and Discussion ...................................................................................... 149
7.3.1 Effect of battery geometry .............................................................................. 149
7.3.2 Influence of the geometrical parameters in battery performance ................... 152
7.3.2.1 Effect of battery dimensions and current collector positions in the horseshoe
geometry .................................................................................................................. 152
7.3.2.1.1 Current collector positions ........................................................................ 153
7.3.2.1.2. Dimensions of the battery ........................................................................ 154
7.3.2.2 Influence of the radius in the ring geometry ................................................ 157
7.3.2.3 Comparative performance of ring and gear battery geometries .................. 158
7.4 Conclusions ........................................................................................................ 163
7.5 References .......................................................................................................... 164
8. Computer simulation of the effect of different thermal conditions in the
performance of conventional and unconventional lithium-ion battery geometries .. 169
8.1 Introduction ........................................................................................................ 171
xx
8.2 Preparation and measurement of the full-cell .................................................... 173
8.3 Theoretical model: parameters, initial values and boundary conditions ........... 174
8.3.1 Theoretical simulation model ......................................................................... 174
8.3.2 Specific parameters and initial values ............................................................ 175
8.3.3 Boundary conditions ....................................................................................... 180
8.4 Results and discussion ....................................................................................... 183
8.4.1 LiC6/LiFePO4 full-cell: Validation of the theoretical model .......................... 183
8.4.2 Battery performance of the various battery geometries at different thermal
conditions ................................................................................................................. 184
8.4.2.1 Isothermal condition .................................................................................... 184
8.4.2.2 Adiabatic condition ...................................................................................... 186
8.4.2.3 Environmental conditions ............................................................................ 188
8.4.3 Total heat at low and high discharge rates ...................................................... 192
8.4.4 Ohmic heat for ring geometry with different radius ....................................... 197
8.5 Conclusions ........................................................................................................ 203
8.6 References .......................................................................................................... 204
9. Conclusions and future work ................................................................................. 209
9.1 Conclusions ........................................................................................................ 211
9.2 Future work ........................................................................................................ 213
xxi
List of figures
Figure 1.1 - Battery evolution with respect to their energy density. ............................... 3
Figure 1.2 - Schematic representation of the main structure of a lithium ion battery and
the process of insertion/extraction of lithium ions that occurs at the electrodes during
discharge of a battery. ....................................................................................................... 7
Figure 2.1 - 1C discharge voltage curve comparison between the rigorous model and
the simplified model at different number of terms or node points through the Galerkin’s
approximation. Figure from [11]. ................................................................................... 32
Figure 2.2 - Cell configuration (not to scale). The x-dimension corresponds to the
length of the cell and the y-dimension corresponds to the height of the cell. Figure
adapted from [24]. .......................................................................................................... 33
Figure 2.3 - Experimental and simulated discharge curves for PLION cells at low
rates. The C rates for thin, medium and thick cells are 2.312, 2.906, and 3.229 mA/cm2,
respectively. The dots represent the experimental data and the solid lines correspond to
the simulation results. Figure from [31]. ........................................................................ 36
Figure 2.4 - Solution phase diffusion coefficient as a function of discharge rate used to
fit experimental data for three different cells. 1C corresponds to 1.156, 1.937 and 2.691
A/m2 for thin, medium and thick cells, respectively. Figure from [6]. .......................... 37
Figure 2.5 - Conduction phenomena in the LiFePO4 cathode during battery charging.
Figure from [32]. ............................................................................................................ 39
Figure 2.6 - Illustration of the composition of the cathode electrode: complementary
solid phase and electrolyte phase [35]. .......................................................................... 42
Figure 2.7 - Schematic computation domain of a Li–air battery during discharge
operation. The inset demonstrates the discharge products formation of Li2O2 and
Li2CO3 covering the porous carbon surface. Figure from [45]. .................................... 44
Figure 2.8 - Temperature on the cell surface during 1C discharge process under
different cooling conditions. Figure from [46]. ............................................................. 46
Figure 2.9 - Cell voltage for 1C discharge process under different cooling conditions.
Figure from [46]. ........................................................................................................... 46
Figure 3.1 - Steps for the implementation of the simulations........................................ 58
Figure 3.2 - Representation of the dimension of the battery for the application of the
theoretical model: a) 1D, b) 2D and c) 3D. .................................................................... 59
Figure 3.3 - Design of different geometries for lithium-ion batteries. .......................... 60
xxii
Figure 3.4 - Different size of the mesh: extremely fine, fine and normal. .................... 61
Figure 4.1 - Schematic representation of the main structure of a lithium ion battery. .. 70
Figure 4.2 - Voltage as a function of delivered capacity at different scan rates for: a)
free electrolyte and b) battery separator membrane with 70% of porosity and 3.8 of
tortuosity. ........................................................................................................................ 76
Figure 4.3 - Delivered capacity as a function of the scan rate for free electrolyte and
separator membrane batteries. ........................................................................................ 77
Figure 4.4 - Voltage as a function of delivered capacity for batteries with separator
membranes with different degrees of porosity with tortuosity of 3.8 at scan rates of a)
0.15C and b) 5C. ............................................................................................................. 78
Figure 4.5 - Delivered capacity as a function of the degree of porosity at different scan
rates: 0.15C, 2C and 5C. ................................................................................................. 79
Figure 4.6 - Delivered capacity as a function of tortuosity for membranes with different
degrees of porosity: a) low scan rate, 0.15C, b) moderate scan rate, 2C and c) high scan
rate, 5C. .......................................................................................................................... 80
Figure 4.7 - Voltage as a function of the delivered capacity for battery separator
membranes with different thicknesses, 70% of porosity and 3.8 of tortuosity: a) 0.15C
and b) 5C. ....................................................................................................................... 82
Figure 4.8 - Delivered capacity as a function of the separator thickness at different scan
rates: 0.15C, 2C and 5C. ................................................................................................. 83
Figure 5.1 - Voltage as a function of the delivered capacity at C/10 and C/2 discharge
rates for the a) Li/LFP and b) Li/LMO half-cells. .......................................................... 97
Figure 5.2 - Voltage as a function of delivered capacity for Li/LFP half-cells with C1:
95% a) and 50% b) at a discharge rate of 1C. ................................................................ 98
Figure 5.3 - Delivered capacity as a function of C3 for different C1 for Li/LFP (a) and
Li/LMO (b) half-cells at a discharge rate of 1C. ............................................................ 99
Figure 5.4 - Minimum percentage of C3 as a function of C1 for both half-cells at a
discharge rate of 1C. ..................................................................................................... 100
Figure 5.5 - Delivered capacity and Capacitysim/Capacitytheo (%) ratio as a function of
C1 for the Li/LFP half-cell at 1C discharge rate. .......................................................... 101
Figure 5.6 - a) Delivered capacity as a function of minimum C3 for the Li/LFP half-
cells: a) C1=95% at 1C, 5C and 10C discharge rates and b) C1 = 95%, 75% and 50% at
5C discharge rate. ......................................................................................................... 102
xxiii
Figure 5.7 - Nyquist plot for the Li/LFP half-cell: a) C1 = 95% with different C3 values
at 1C discharge rate and b) C1 = 50% with different C3 values at 1C discharge rate.
Nyquist plot for Li/LMO half-cells: c) C1 = 95% and 50% and C3 = 1% and 10% at 1C
discharge rate. ............................................................................................................... 105
Figure 5.8 - Total impedance as a function of minimum C3 for different C1 at 1C
discharge rate for: a) Li/LFP and b) Li/LMO half-cells. .............................................. 106
Figure 5.9 - Schematic representation of a battery cathode and the corresponding
intercalation process during the discharge mechanism. ............................................... 107
Figure 5.10 - Electrolyte and electrode current density as a function of cathode length
for a Li/LFP half-cell with C1 = 95% and C3 = 4% of at 1C discharge rate and at 500s.
The blue line corresponds the sum of both current densities along the width of the
cathode, showing that the divergence of the total electric charge is null. .................... 108
Figure 5.11 - Electrolyte current density as a function of the cathode length for Li/LFP
half-cell for various C3 at 1C discharge rate and at 500s for C1= 95% (a) and 50% (b).
...................................................................................................................................... 109
Figure 5.12 - Electrode current density as a function of cathode length for Li/LFP half-
cell for various C3 at 1C discharge rate and 500s for C1 = 95% (a) and 50% (b). ........ 110
Figure 5.13 - Electrolyte and electrode current density as a function of time for a
Li/LFP half-cell with C1 = 95% and C3 = 0.9% at 20 µm of position inside of cathode in
relation to separator/cathode interface. The width of the cathode is 70 µm. ................ 110
Figure 6.1 - Schematic representation of a conventional (a) and an interdigitated (b)
battery with indication of the main geometrical parameters. ....................................... 120
Figure 6.2 - Schematic representation illustrating how the area of each component is
maintained constant, while varying the number of digits. ............................................ 122
Figure 6.3 - Delivered capacity at 1C discharge rate as a function of the anode thickness
for a fixed cathode thickness of 400 μm (a) and as a function of the cathode thickness
for a fixed anode thickness of 200 μm (b). ................................................................... 125
Figure 6.4 - Delivered capacity as a function of the scan rate for three different anode
thicknesses and fixed cathode thickness of 400 μm. .................................................... 126
Figure 6.5 - Delivered capacity as a function of the scan rate (a and c) and number of
digits (b). Separator thickness and battery width as a function of the number of digits
with a fixed c_dig at 400 μm and e_dig at 20 μm (d). .................................................. 128
Figure 6.6 - Nyquist plot for the conventional (a) and the interdigitated (b) geometry
with 8 digits in frequency range of 1 mHz to 1MHz. ................................................... 129
xxiv
Figure 6.7 - a) Delivered capacity and b) width of the battery as a function of digit
length for a four digits battery for a constant (I) and a variable (II) separator. ............ 131
Figure 6.8 - Nyquist plot of interdigitated geometries for three different digit lengths in
the frequency range from 1 mHz to 1MHz................................................................... 132
Figure 6.9 - a) Delivered capacity and b) width of the battery as a function of digit
thickness for a constant (I) and a variable (II) separator. ............................................. 132
Figure 6.10 - Nyquist plot of the interdigitated geometries for three different digit
thicknesses in the frequency range from 1 mHz to 1MHz. .......................................... 133
Figure 6.11 - Schematic representation of the: a) digit limit length and b) digit limit
thickness for four digits. ............................................................................................... 134
Figure 6.12 - Delivered capacity as a function of digit limit thickness (a) and length (b)
at 200C and 400C. c) Width of the battery as a function of the number of digits for
c_dig= 100 m and e_dig=20 m at 200C and 400C. ................................................. 135
Figure 7.1 – Delivered capacity as a function of scan rate for the different batteries. 149
Figure 7.2 - Delivered capacity for the different geometries as a function of a)
maximum distance and b) distance between collectors. ............................................... 150
Figure 7.3 - a) Schematic representation of the current collector positions and b)
voltage as a function of the delivered capacity for the different current collector
positions. ....................................................................................................................... 153
Figure 7.4 - Delivered capacity as a function of current collector positions and
maximum distance of lithium ions. .............................................................................. 154
Figure 7.5 - a) Schematic representation of the horseshoe battery dimension, L_dim,
and b) delivered capacity as a function of L_dim. ....................................................... 155
Figure 7.6 - Maximum distance and distance between current collectors as a function of
L_dim for the horseshoe geometry. .............................................................................. 156
Figure 7.7 - a) Schematic representation of the ring geometry and b) delivered capacity
as a function of the radius, Rd. ...................................................................................... 157
Figure 7.8 - Maximum distance, distance between current collectors and thickness of
the separator as a function of Rd. .................................................................................. 158
Figure 7.9 – Schematic representation of the gear geometry. ..................................... 159
Figure 7.10 – Voltage as a function of the delivered capacity for the ring and gear
geometries with different Rg: a) 93.9 µm and b) 20 µm. ............................................. 160
Figure 7.11 - Electrolyte potential and electrolyte current density vectors for a) ring and
b) gear geometries. ....................................................................................................... 161
xxv
Figure 7.12 - Voltage as a function of the delivered capacity for the ring and gear
geometries with different separator thickness. ............................................................. 162
Figure 8.1 - Schematic representation of the boundary conditions applied in the
conventional geometry. ................................................................................................ 182
Figure 8.2 - Voltage as a function of the delivered capacity at C/10 rate for the
LiC6/LiFePO4 full-cell with a conventional geometry. ................................................ 183
Figure 8.3 - Delivered capacity as a function of scan rate for all geometries under
isothermal condition. .................................................................................................... 185
Figure 8.4 - Delivered capacity (a) and temperature (b) as a function of the scan rate for
all geometries under adiabatic condition. ..................................................................... 186
Figure 8.5 - Nyquist plot for conventional and interdigitated geometries under adiabatic
condition. ...................................................................................................................... 188
Figure 8.6 - Delivered capacity (left) and final temperature (right) as a function of the
scan rate for all geometries under cold (a and b), regular (c and d) and hot (e and f)
conditions. .................................................................................................................... 190
Figure 8.7 - Total heat in the anode (a), separator (b) and cathode (c) for all geometries
at 1C as a function of the time. d) Total heat along the battery for all geometries at 1C
after 120 000s. .............................................................................................................. 192
Figure 8.8 - Temperature of the battery as a function of time for all geometries at 1C.
...................................................................................................................................... 194
Figure 8.9 - Total heat for anode (a), separator (b) and cathode (c) for all geometries at
300C as a function of time. ........................................................................................... 195
Figure 8.10 - Total heat along the battery after 50 s at 300C for conventional and
interdigitated geometries (a) and for the remaining geometries (b). ............................ 196
Figure 8.11 - Temperature as a function of time for all geometries at 300C. ............. 197
Figure 8.12 - Schematic representation of the ring geometry for the radius of 93.9 µm
and 430 µm. .................................................................................................................. 198
Figure 8.13 - a) Capacity as a function of ring radius and b) temperature as a function
of time for all ring radius at 500 C. .............................................................................. 198
Figure 8.14 - Ohmic heat for anode (a), separator (b) and cathode (c) as a function of
the time at 500 C for various ring radius. ..................................................................... 199
Figure 8.15 - Ohmic heat along different places between the current collectors of the
battery after 70 s at 500C for ring geometry with different radius. .............................. 200
Figure 8.16 - Nyquist plot for the ring geometry with different radius at 500 C. ....... 201
xxvi
Figure 8.17 - Ionic current density vectors of the ring geometry for a) R= 93.9 µm and
b) R=430 µm. ............................................................................................................... 202
xxvii
List of tables
Table 1.1 - Main advantages and disadvantages of lithium ion batteries when compared
to related battery systems [33–36]. ................................................................................... 6
Table 1.2 - Nomenclature adopted for the variables of the mathematical models. ....... 11
Table 1.3 - Summary of the main equations governing the different processes involved
in lithium-ion batteries. .................................................................................................. 13
Table 1.4 - Summary of the boundary conditions or limits of the mathematical model
adopted by [110] where La, Ls and Lc are the width of the anode, separator and
cathode, respectively. ..................................................................................................... 16
Table 3.1 - Equations governing various phenomena within a battery [1-4]................. 62
Table 4.1 - Boundary conditions applied in the simulation. The nomenclature is
indicated in the List of Symbols and Abbreviations. ..................................................... 72
Table 4.2 - Parameters used in the simulations.............................................................. 75
Table 4.3 - Limit value of tortuosity for different degrees of porosity and scan rates. . 81
Table 5.1 - Parameters used for the simulations of the Li/LFP and Li/LMO half-cells..
........................................................................................................................................ 95
Table 5.2 - Minimum values of n=C2/C3 as a function of C1 for the Li/LFP and Li/LMO
half-cells at a discharge rate of 1C. .............................................................................. 101
Table 5.3 - Minimum values of the n ratio for different C1 for Li/LFP half-cells at 1C,
5C and 10C discharge rates. ......................................................................................... 103
Table 6.1 - Parameters used in the simulations of the conventional and interdigitated
battery structures........................................................................................................... 123
Table 7.1 - Parameters used for the simulations, main characteristics and applications
for the different battery geometries [44-46]. ................................................................ 146
Table 8.1 - Values of the parameter values used in the simulations. The nomenclature is
indicated in the List of Symbols and Abbreviations. ................................................... 175
Table 8.2 - Schematic representation of the different battery geometries and the
corresponding dimensions. The nomenclature is indicated in the List of Symbols and
Abbreviations. .............................................................................................................. 177
Table 8.3 - Summary of the boundary conditions implemented in the conventional
geometry. The nomenclature is indicated in the List of Symbols and Abbreviations. . 181
1.Introduction
1
1. Introduction
This chapter is divided into three parts: the theme of the thesis is introduced, the
main objectives are presented as well as the thesis structure and the applied
methodology.
With respect to the introduction, it is shown the importance of lithium-ion batteries
as energy storage systems, the mathematical models for lithium ion batteries, the
description of the main materials used for each of the components of a battery (anode,
cathode and separator) and how material characteristics affect battery performance.
Finally, it is introduced the general mathematical framework for the microscopic models
of lithium-ion batteries.
This chapter is partially based on the following publication:
“Lithium ion rechargeable batteries: State of the art and future needs of
microscopic theoretical models and simulations”, D. Miranda, C.M. Costa, S.
Lanceros-Mendez, Journal of Electroanalytical Chemistry 739 (2015) 97-110.
1.Introduction
3
1.1 Introduction
The XX and XXI centuries are characterized by rapid technological advances, in
particular in the electronics, informatics and communication industries. The
development of products such as computers, mobile phones, tablets and other portable
devices lead to an increasing need for battery autonomy and performance [1–3].
Increasing battery performance (Figure 1) is associated to the use of novel materials
and concepts leading to increasing loading capacity, cycle life and safety [4–7]. Figure
1.1 illustrates the evolution of batteries with respect to energy density.
Nowadays, large attention is being paid to the development of batteries for the
automobile industry in order to reduce fossil fuel dependence and emission gases
responsible for the greenhouse effect and therefore to reduce the environmental impact
associated to the energies used for mobility [8–10].
Figure 1.1 - Battery evolution with respect to their energy density.
The main goal of the battery industry is to obtain specific levels of battery
performance for the different applications (e.g. applied voltage and capacity) with low
production costs. In this context, intensive research is being devoted to the development
of rechargeable or secondary batteries [11,12].
1.Introduction
4
For many years, nickel–cadmium batteries (Ni–Cd) were the most suitable for
portable communications systems and computing equipments. However, at the
beginning of the 90s, lithium ion batteries increased in attention and acceptance by
consumers. Nowadays, lithium ion batteries are the most widely used and still show a
promising growth potential [13,14]. The pioneering work with lithium ion batteries
began in 1912 and it was in the 70s that the first non-rechargeable lithium ion batteries
were commercialized [15,16]. Lithium is the lightest of all metals, showing a large
electrochemical potential and high energy density relative to its weight [17]. Several
attempts to develop rechargeable lithium ion batteries failed due to safety problems
[18,19], associated to the inherent instability of lithium metal, in particular during the
charge cycle.
The lithium ion is safe provided that certain precautions are taken during battery
charge and discharge cycles. The safety of the lithium-ion battery is one of the key
issues for improving the performance of the battery. Thus, the interest in developing
lithium-ion batteries increased and in 1991 the Sony Corporation commercialized the
first lithium-ion batteries [20].
For increasing battery performance and optimizing materials and designs it is
critical to have suitable theoretical models that allow battery simulation. The
mathematical theoretical models for lithium-ion batteries describe the physical
processes and mechanisms of the different components of the batteries and are essential
for optimizing performance, design, durability and safety of lithium-ion batteries.
Mathematical models for lithium ion batteries have been developed at different
scales of battery operation from the macro to the nano scales [21].
The mathematical models at the micro-scale are the most widely used for research,
development and battery optimization as they allow the correlation of the theoretical
results with experimental transport and electrochemistry data [22].
This review is divided into the following sections: first, the advantages and
disadvantages of lithium-ion batteries in relation to other types of batteries are outlined;
then, the process of insertion/extraction of lithium ions and each of the main
components of the battery are described; finally, the microscopic mathematical models
dealing with the description of the operation of lithium ion batteries are reviewed and
their results discussed; the reviews finish with some concluding remarks on the open
questions and future research directions in this specific topic.
1.Introduction
5
1.1.2 Advantages and disadvantages of lithium ion batteries
A critical assessment on the main advantages and disadvantages should be
performed for each type of battery [2]. The main advantages and disadvantages of the
use of lithium ion batteries when com- pared to other types of batteries such as Ni–Cd,
Lead–Acid battery and Nickel–Metal Hydride Cells are illustrated in Table 1. By com-
paring lithium ion batteries with nickel–cadmium batteries (Ni– Cd), the energy density
of the lithium ion batteries is approximately twice as large as the energy density of
nickel–cadmium batteries [23,24]. The charging cycle, on the other hand, shows similar
characteristics for nickel–cadmium and lithium-ion batteries [25,26]. Lithium-ion
electrochemical cells show high voltages and in case, for example, of an electrical
apparatus requiring a voltage of 3.6 V, it requires just one cell instead of a package of
three cells of 1.2 V for nickel–cadmium batteries. Lithium-ion batteries show no
memory effect in their charge and discharge cycles which leads to increased life time
[27]. Furthermore, their self-discharge effect is lower in comparison to nickel–cadmium
batteries. Despite the mentioned advantages, lithium-ion batteries also show some
disadvantages. In particular, lithium-ion batteries require a protection circuit to maintain
safe operation. This protection circuit limits the peak voltage of each cell during charge
and prevents the cell voltage to strongly decrease during discharge [28].
The temperature of lithium-ion batteries should be also con- trolled in order not to
exceed 100 °C. The maximum charge and discharge current in the majority of these
batteries is limited between 1C and 2C [29]. Aging is also a concern for most lithium-
ion batteries and deterioration is observed after one year, approximately, whether in use
or not [30–32]. However, in some specific applications the durability of lithium-ion
batteries can extend up to about five years [7].
1.Introduction
6
Table 1.1 - Main advantages and disadvantages of lithium ion batteries when compared
to related battery systems [33–36].
Advantages Disadvantages
High energy density, between 100 and
150 Wh kg-1
A protection circuit is needed for
maintaining constant voltage
One regular charge cycle is needed,
not needing a long charging cycle.
Subject to aging, while not in use.
Low self-discharge when compared
with Ni–Cd batteries.
Restrictions on transportation.
Transportation of large quantities may be
subjected to regulatory control.
Low maintenance and no memory
effects.
High manufacturing costs due to the
price of lithium.
Specific cells can provide high current
for particular applications.
Lithium batteries show good operation
range for discharge currents between 1C
and 2C.
In the automotive industry there are many options for electric vehicle batteries, each
system offering unique features with advantages and disadvantages [37–39]. Currently,
some of the most promising approaches are based on lithium-ion batteries, due to their
high energy density [7]. However, lithium-ion batteries show problems with sensitivity
of overload that can reduce their life cycle. Other options under consideration include
fuel cells with rechargeable batteries. In any case, it should be noted that these options
do not provide the same amount of energy in comparison to fossil fuels: ~40 MJ/kg for
fossil fuel against 1.5–0.25 MJ/kg for fuel cells and advanced batteries, respectively
[34]. Although electric vehicles are being designed and built, currently there is no
energy source that matches the power and energy of the internal combustion engine
[40,41]. Nevertheless, research is conducted to develop a robust system capable of
achieving reasonable acceleration for the vehicle and the ability to perform long
distances [42]. In this sense, fuel cells and lithium-ion batteries are suitable alternatives
for application in electric vehicles due to their large improvement potential based on
novel materials and optimized design [43–45].
1.Introduction
7
1.1.3 Mathematical model for lithium ion batteries
Lithium ion batteries are composed by three major components (Figure 1.2): anode,
cathode and separator [46,47].
As in other types of batteries, it shows two electrodes with different electrical
potentials related to the chemical nature of their active material which are the cathode
and the anode [48–50]. The battery separator is located between the cathode and the
anode and it is an ionic conductor but electronic insulator. Lithium ion batteries also
need the electrolyte, which may be embedded in the separator, containing lithium salts
dissolved in an organic solvent and that can be dispersed in the three battery
components (electrodes and separator) as illustrated in Figure 1.2 [51,52].
Figure 1.2 - Schematic representation of the main structure of a lithium ion battery and
the process of insertion/extraction of lithium ions that occurs at the electrodes during
discharge of a battery.
The operation of a lithium-ion battery is based on a process called ‘‘rocking chair’’
due to the extraction and insertion of lithium ions at the electrodes. During discharge of
a battery, extraction of lithium ions from the anode occurs, providing electrons to the
cathode through an external circuit (Figure 1.2). When the lithium ions reach the
cathode, capture of electrons from the external circuit occurs together with the insertion
process of lithium ions. During the discharge process, therefore, electrons and Li-ions
move from the anode to the cathode.
The insertion/extraction process has advantages and disadvantages when compared
to the others traditional battery processes (such as the ones in Ni–Cd) [53]. Insertion and
1.Introduction
8
extraction processes are highly reversible but, on the other hand, they are associated to a
change in the volume of the electrodes that, depending on their nature, leads to matrix
degradation over the lifecycle [54].
The majority of the theoretical models consider that the active material in both
electrodes is spherical and that it is supported by a material that is not involved in the
battery operation reactions, i.e., an inert material.
The process of insertion/extraction of lithium ions and the over- all battery
operation can be studied from different points of view and at different physical and
chemical scales: nanoscale, mesoscale, microscale and macroscale, as illustrated in
Figure 3. Two recent reviews [21,55] describe the theoretical simulations for anode,
cathode and separator as well as the interface between electrodes and electrolyte
considering a nano- and meso scale approach, mainly focusing on the ion transport
phenomena at the meso scale.
For all models developed at the different physico-chemical scales, there are a
number of variables available for manipulation, particularly relevant for battery
performance. The electrodes, for example, are studied taking into account different
scales and particle shapes, among others, based on computer-aided reconstruction.
Effects of mechanical stress and thermal heterogeneities are also studied from the
atomic to the macroscopic scales.
The development of models at different scales (multi-scale approach) are suggested
in order to prove battery operation coupling at different physical levels [21], as nano-
and mesoscale models are suitable for understanding and improving the different
components of the battery from a materials science point of view but lack for proper
validation with respect to improvement in batteries performance. Suitable extrapolation
from the lower to the higher scales are needed in order to achieve the final goal, which
is to allow proper battery design.
Thus, a more detailed physico-chemical description of the materials is necessary for
improving battery design optimization by increasing predictability of multiscale models
[55].
On the other hand, the purpose of mathematical models at the microscopic level is
to study parameters directly affecting battery performance, such as energy density,
capacity, voltage and discharge time which are readily modeled considering overall
properties of different materials, their microstructures, electrolytes and boundary
conditions.
1.Introduction
9
1.1.4 Materials
1.1.4.1 Anode and cathode electrodes
The electrodes in most batteries are porous [56], although in some cases may be
compact and flat. Several materials have been used for electrodes, the most frequently
used being graphite as anode material [57–59] and LiCoO2, LiNiO2, LiMn2O4,
LiMn1/3Ni1/3 Co1/3O2 and LiFePO4 as cathodes [60–63]. The most promising cathode
materials are from the LiMPO4 family in which phosphorous occupies tetrahedral sites
and the transition metal (M) occupies octahedral sites. In this family, the most used
cathode material is lithium iron phosphate (LiFePO4), which shows high open circuit
voltages >3.5V but low capacities around ~170mAhg-1 [64].
At the present moment, the most commonly used cathode material in lithium-ion
batteries for portable applications is LiCoO2 [65], but cobalt is less available and shows
a higher price than other transition metals. The cost of LiNiO2 is lower and shows
higher energy density but is less stable and has a less ordered structure when compared
with LiCoO2 [65]. In this sense, cathodes with different amounts of three transition
metals Li(Ni, Mn, Co)O2 are increasingly being used as they show high capacity, good
rate capability and can operate at high voltages [60,65]. The main characteristics of the
materials used for cathode development are the presence of a transition metal ion for
maximizing cell voltage, the possibility of preparation of a composite with the active
material to allow insertion/extraction of a large quantity of lithium ions for maximizing
the capacity of the cell and, finally, the composite material must possess minimal
structural changes depending on the composition of lithium, which ensures good
reversibility of the process. Relatively of the anode material, graphite improves the
insertion or intercalation, being able to store lithium through the interstitial sites
between two graphite planes. This process is directly related to the energy storage
density of Li-ion batteries. Graphite also shows low expansion, which is directly related
to their facility to maintain their charge capacity after many charge–discharge cycles.
Further, it is cheap, shows cycle efficiency and moderate capacity, 373 mAhg-1 [66].
Carbon nanotubes (CNT) are also used as anode material. Single walled CNT show
higher capacity, up to 1000 mAhg-1, than graphite and can be used as a support matrix.
Finally, they have adequate properties for electrode materials such as high tensile
strength and high conductivity. The disadvantage of CNT is their irreversible lithium
ion capacity loss that occurs during the first cycle. [67].
1.Introduction
10
1.1.4.2 Battery separator
The separator is a key component in all electrochemical devices and is located
between the anode and the cathode [68,69]. The role of the separators is to serve as the
medium for the transfer of the lithium ions between both electrodes and to control
lithium ion flow and mobility [70]. The key requirements of a separators for lithium ion
batteries are thickness, permeability, gurley, porosity and pore size, wettability by liquid
electrolyte, electrolyte absorption and retention, resistance to chemical degradation by
electrolyte impurities, dimensional stability, puncture strength, thermal stability,
mechanical and dimensional stability and skew [68,71].
The separator membrane is often a polymer matrix, in which the membrane is
impregnated by the electrolyte solution. The liquid electrolyte solution is constituted by
salts dissolved in solvents, water or organic molecules. The solvent must meet the
requirements of low viscosity, medium to high dielectric constant for dissolving the
salts, low viscosity for facility the ion transportation, to be inert to all cell components
and remain in liquid state in the temperature range of cell operation cell [72,73]. The
most used solvents in electrolyte solutions are ethylene carbonate (EC), propylene
carbonate (PC), dimethyl carbonate (DMC), diethyl carbonate (DEC) and ethyl methyl
carbonate (EMC) [74–77].
The lithium salts most used in electrolyte solution are Li(CF3SO2)2N [78], LiAsF6
[79], LiPF6 [80], LiClO4 [81], LiBF4 [82], LiCF3SO3 [83] in which the size of the
anions is an important factor that determines the properties of the salts [84].
The materials used as separator materials are polymers with/ without dispersed
fillers. Among the used polymers stand out poly(ethylene) (PE) [85], poly(propylene)
(PP) [86], poly(ethylene oxide) (PEO) [87,88], poly(acrylonitrile) (PAN) [88,89] and
poly (vinylidene fluoride), PVDF, and its copolymers [90–92] (poly(vinylidene-co-
trifluoroethylene), PVDF-TrFE [93], poly(vinylidene fluoride-co-hexafluoropropene),
PVDF-HFP [94] and poly(vinylidenefluoride-co-chlorotrifluoroethylene), PVDF-CTFE.
PVDF and copolymers show important advantages in comparison to polyolefins and
other materials for their use as separators due to their polarity (high dipole moment) and
high dielectric constant for a polymer, which can assist the ionization of lithium salts. It
is possible to control their porosity, they are wetted by organic sol- vents and are
chemically inert. They also show good contact between electrode and electrolyte and
are stable in cathodic environment [95]. The fillers incorporated (dispersed directly) into
1.Introduction
11
the polymer hosts may be inert oxide ceramic (Al2O3, SiO2, TiO2), molecular sieves
(zeolites), ferroelectric materials (BaTiO3) and carbonaceous fillers, among others, with
the goal to increase the electrochemical properties, mechanical and thermal stability of
the separator [96].
1.1.5 General mathematical framework for the microscopic models of lithium-ion
batteries
Most mathematical models for lithium ion batteries are developed to study the
performance of the battery in one and two-dimensions by considering electrochemical
and transport processes in the different components of the battery. Some models also
allow to study of the influence of temperature in the performance of the battery [97].
The different microscopic models are based on the Doyle/Fuller/Newman model [98–
109], considering the same mathematical framework for the electrochemical phenomena
and transport occurring in the different components of the battery: anode, cathode and
separator with electrolyte. The main differences between the developed theoretical
studies are thus reduced to border and boundary conditions, specific for each of the
studies, which simplify the general mathematical model for each particular case under
study [98–109] and/or in the materials used for the different components. In the
following, the nomenclature adopted for the variables of the mathematical models is
introduced in Table 1.2.
Table 1.2 - Nomenclature adopted for the variables of the mathematical models.
Nomenclature
a specific interfacial area, m2/m3
CL concentration of Li ions in the electrolyte, mol/m3
CE concentration of Li ions in the electrode, mol/m3
D diffusion coefficient of the salt in the electrolyte, m2/s
DLI diffusion coefficient of Li ions in the electrode, m2/s
F faraday’s constant, 96487 C/mol
f activity of the salt in the electrolyte, mol/m3
iE current density in the electrode, A/m2
iL current density in the electrolyte phase, A/m2
ITOTAL total current density, A/m2
1.Introduction
12
jLi+ pore wall flux of Li ions, mol/cm2 s
L width
M mass transport flux, mol/m2
R reaction term of the mass balance equation, mol/m3 s
R gas constant, 8,314 J/mol K
Rf film resistance, m2
r radius of electrode spherical particle, m
T temperature
t time, s
0
t transport number of the positive ion
0u open circuit voltage, V
i porosity of region i(i = a,s,c)
f,i volume fraction of fillers in electrode i ( i = a,s,c)
over-potential, V
E potential of the electrodes, V
L potential of electrolyte,V
ionic conductivity of the electrolyte, S/m
ef effective ionic conductivity of the electrolyte, S/m
electronic conductivity of the solid phase of the electrode i (i = a,s,c), S/m
ef,i effective electronic conductivity of the solid phase of the electrode i (i =
a,s,c), S/m
Subscripts
a anode
c cathode
s separator
0 initial condition
Thus, the general mathematical model presented in this review is based on the
Doyle/Fuller/Newman model which describes the fundamental equations governing the
main phenomena that occur in the operation process of a lithium-ion battery. The main
equations governing the different processes during operation of a battery are presented
in Table 1.3.
1.Introduction
13
Table 1.3 - Summary of the main equations governing the different processes involved
in lithium-ion batteries.
Cathode
Governing Equation Description
Li
LcefLc jta
x
CD
t
C)1( 0
2
2
2
,
Diffusion of lithium ions in the
electrolyte applied to the cathode.
Li
EcefFaj
x 2
2
2
,
Electrode potential calculated
by the Ohm Law where the current
density gradient is substituted by its
equivalent in terms of lithium ion
flux according to Faraday’s Laws.
2
0
2
2
2
, ln)1(
2
x
Ct
F
kRTFaj
x
kL
Li
Lcef
This equation relates the
potential of the electrolyte with the
local current density in the cathode
(Ohm Law).
Anode
Governing Equation Description
Li
LaefLa jta
x
CD
t
C)1( 0
2
2
2
,
Diffusion of lithium ions in the
electrolyte applied to the anode.
Li
EaefFaj
x 2
2
2
,
Electrode potential calculated
by the Ohm Law where the current
density gradient is substituted by its
equivalent in terms of lithium ion
flux according to Faraday’s Laws.
2
0
2
2
2
, ln)1(
2
x
Ct
F
kRTFaj
x
kL
Li
Laef
This equation relates the
potential of the electrolyte with the
local current density in the anode
(Ohm Law).
Electrolyte/
Separator
Governing Equation
Description
2
2
2
,
x
CD
t
C LsefLs
Lithium ion diffusion in the
electrolyte.
2
0
2
2
2
, ln)1(
2
x
Ct
F
kRTFaj
x
kL
Li
Lsef
This equation relates the
potential of the electrolyte with the
local current density (Ohm Law).
1.Introduction
14
Active
material
Governing Equation
Description
r
C
rr
CD
t
C EELi
E 22
2
Diffusion of lithium ions in the
active material.
General
equations
Governing Equation
Description
Faraday's law
(electrodes)
LiL Faji
Faraday’s law express the
relationship between the
insertion/extraction of lithium ions
into the electrodes with the
electrical charge flow
Relation between the lithium
ions flux and the current density in
the electrodes.
Faraday's law
(electrolyte)
LiE Faji
Relation between the lithium
ions flux and the current density in
the electrolyte (Faraday ́ s Law).
Total current
density TOTALLE Iii
Conservation of charge. The
current density is preserved
between the electrode and the
electrolyte.
Butler-Volmer
equation
(kinetics)
0002
exp2
expuRT
F
uRT
Fijn
Kinetics of the heterogeneous
reaction at the electrode/electrolyte
interface, described by the Butler–
Volmer equation.
Variable over-
potential
0uLE
The variable over-potential
relates the potential of the
electrodes/electrolyte and the open
circuit voltage.
Mass transport
process F
tiCL
C
CDM L
L
LTotal0
ln
ln1 0
Mass transport flux.
Term of the
reaction
Li
jtv
aR )1( 0
Reaction term of the mass
balance equation.
1.Introduction
15
Overall mass
balance RM
t
CL
.
Overall mass balance.
Auxiliary
equations
Governing Equation
TOTALE
LLL
LL
LiIdxFaj
Csa
sa
0,
TOTALL
L
LiIdxFaj
s
0,
0
ascicc
ccikef
,,)106018.110509.1
107212.410007.5101253.4(,
414310
2742
)1( ,, ifiiief
aciDD brugg
iiief ,,,
The different models found in the literature involve simplifications and specific
boundary conditions of the previous equations in order to account for specific
phenomena [98–109]. The auxiliary equations are important as they reflect the effect of
the microstructure (porosity) in the ionic conductivity and diffusion process in all
components of the battery. These effects are described through the Bruggeman equation
for highly conductive isotropic materials [106]. This equation applies to the ionic
conductivity in liquid-electrolyte-soaked porous media, not being suitable for electronic
conductivity based on networks of touching particles or for solid polymer composite
electrolytes [106].
The model of the battery in one dimension is considered taking into account the
three main components of the battery (separator, anode and cathode) in dimension x and
sub-dimension r (spherical particles of active material). In the following, the boundary
conditions adopted for the different equations at the interfaces between the regions will
be presented.
The diffusion of lithium ions in the electrolyte occurs at the three cell components
(anode, cathode and separator). The collectors of the battery are a wall impermeable to
the electrolyte, so the flow of lithium ions is null at these limits. The interfaces of the
three components show a condition of continuity that is expressed as an equal mass that
flows on both sides of the interface, i.e.
1.Introduction
16
- at the interfaces:
aa Lx
L
Lx
L
x
C
x
C
(1)
SaSa LLx
L
LLx
L
x
C
x
C
(2)
- at the current collectors:
00
x
L
x
C (3)
0
csa LLLx
L
x
C (4)
• Diffusion of lithium ions in the active material:
00
r
E
r
C (5)
Li
Li
Rr
E
D
j
r
C
sp
(6)
Table 1.4 summarizes the boundary conditions or limits adopted in the model.
Table 1.4 - Summary of the boundary conditions or limits of the mathematical model
adopted by [110] where La, Ls and Lc are the width of the anode, separator and
cathode, respectively.
Region
battery Equation x = 0 x = La x = La + LS x = La+ LS+ Lc
Electrolyte
Li+ diffusion 0
x
CL
Continuity Continuity 0
x
CL
Ohm’s law 0,LL
Continuity Continuity 0
x
L
Electrodes
Ohm’s law 0E
0
x
E 0,EE
TOTALE I
x
Li+ diffusion
0
0
r
C
r
E
Li
LiE
s
D
j
r
C
Rr
1.Introduction
17
1.2 Objectives
The main objective of the present work is the optimization of the performance of
lithium-ion batteries through of computer simulations. This optimization is performed
through the development of theoretical simulations for separators, electrodes and
full/half-cells of lithium ion batteries. For this purpose, the understanding of the main
processes that affect the battery performance is critical, and may be achieved through
adequate simulation based on optimization of electrodes, separators and battery
geometry.
The main specific objectives of this work are:
1) Optimizing the performance of the separator (porous membrane) of lithium ion
batteries through the evaluation of the influence of geometrical parameters such as
degree of porosity, tortuosity, Bruggeman coefficient and thickness. Understand the
relationship of Bruggeman coefficient with the degree of porosity and tortuosity.
2) Obtain the optimal relationship between active material, binder and conductive
additive for lithium-ion battery cathodes. Evaluate the effect of the relative percentages
of active material, binder and conductive additive in cathodes with different active
materials, such as LiMn2O4 and LiFePO4.
3) Evaluate the effect of the geometrical parameters of interdigitated batteries, including
the number, thickness and the length of the digits, on the delivered battery capacity.
4) Study the influence of the geometry in the performance of conventional and
unconventional lithium-ion batteries. Develop new high performance battery geometries
for different applications.
5) Understand the thermal behavior in unconventional geometries for lithium-ion
batteries. Evaluate the heat produced by the different geometries and test the
performance of these batteries at different temperatures and thermal conditions
(isothermal, adiabatic, cold, regular and hot conditions).
1.Introduction
18
1.3 Thesis structure and methodology
The present thesis is divided into nine chapters showing the evolution of the work
during this investigation.
Six of those chapters are based on published or submitted scientific articles.
Chapter 1 presents the introduction to the theme of this thesis, describes the main
objectives of the work and presents the thesis structure and methodology.
Chapter 2 shows the state of the art on the theoretical models for the simulation of
the performance of lithium ion batteries and shows a description of the main theoretical
studies describing the operation and performance of a battery. This chapter also presents
the objectives of the study as well as the structure of the document.
Chapter 3 describes the methodology implemented in the simulations developed in
the different studies. It is also shown the theoretical models used in the different
simulations, such as the electrochemical and thermal models.
The effect of geometrical parameters of the separator, such as degree of porosity,
tortuosity and thickness, in the performance of lithium-ion batteries is presented in
chapter 4. This chapter also shows the relation of Bruggeman coefficient (applied in
equations of ionic diffusion/conductivity) with the degree of porosity and tortuosity of
separator.
Chapter 5 reports the optimal relationship between active material, binder and
conductive additive for lithium-ion battery cathodes. The effect of different percentages
of active material, binder and conductive additive on the performance of two cathodes
with different active materials (LiMn2O4 and LiFePO4) is presented.
The effect of the geometrical parameters of interdigitated batteries, including the
number, thickness and the length of the digits, on the delivered battery capacity is
presented in chapter 6.
The influence of geometry in the performance of conventional and unconventional
lithium-ion batteries is provided in chapter 7. In order to optimize battery performance,
different geometries have been evaluated taking into account their suitability for
different applications, as presented in chapter 7.
The thermal behavior of conventional and unconventional lithium-ion battery
geometries is evaluated in chapter 8. The performance of different battery geometries in
several thermal conditions (isothermal, adiabatic, cold, regular and hot conditions) is
presented.
1.Introduction
19
Finally, chapter 9 provides the general conclusions as well as suggestions for future
work.
1.Introduction
20
1.4 References
1. Crompton, T.P.J., Battery Reference Book2000: Elsevier Science.
2. Munshi, Z.A., Handbook of Solid State Batteries & Capacitors1995: World
Scientific Pub.
3. Wakihara, M. and O. Yamamoto, Lithium ion batteries: fundamentals and
performance1998: Kodansha.
4. Bates, J.B., et al., Thin-film lithium and lithium-ion batteries. Solid State Ionics,
2000. 135(1–4): p. 33-45.
5. Megahed, S. and B. Scrosati, Lithium-ion rechargeable batteries. Journal of
Power Sources, 1994. 51(1–2): p. 79-104.
6. Scrosati, B., Recent advances in lithium ion battery materials. Electrochimica
Acta, 2000. 45(15–16): p. 2461-2466.
7. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
8. Dincer, I., Renewable energy and sustainable development: a crucial review.
Renewable and Sustainable Energy Reviews, 2000. 4(2): p. 157-175.
9. Hill, J., et al., Environmental, economic, and energetic costs and benefits of
biodiesel and ethanol biofuels. Proceedings of the National Academy of
Sciences, 2006. 103(30): p. 11206-11210.
10. Johansson, T.B. and L. Burnham, Renewable energy: sources for fuels and
electricity1993: ISLAND Press.
11. Products, G.E., Rechargeable Batteries Applications Handbook1998: Elsevier
Science.
12. Graham, R.W., Rechargeable batteries: advances since 19771980: Noyes Data
Corp.
13. Consulting, P.R., Advanced Batteries for Portable Power Applications, 2011.
14. Nagaura, T. and K. Tozawa, Progress in Batteries and Solar Cells, 1990. 209.
15. Ozawa, K., Lithium-ion rechargeable batteries with LiCoO2 and carbon
electrodes: the LiCoO2/C system. Solid State Ionics, 1994. 69(3–4): p. 212-221.
16. Colin A, V., Lithium batteries: a 50-year perspective, 1959–2009. Solid State
Ionics, 2000. 134(1-2): p. 159-167.
17. Sapse, A.M. and P. von R. Schleyer, Lithium Chemistry: A Theoretical and
Experimental Overview1995: Wiley.
1.Introduction
21
18. Dusastre, V., J.M. Tarascon, and M. Grätzel, Materials for Sustainable Energy:
A Collection of Peer-Reviewed Research and Review Articles from Nature
Publishing Group2010: World Scientific.
19. Balakrishnan, P.G., R. Ramesh, and T. Prem Kumar, Safety mechanisms in
lithium-ion batteries. Journal of Power Sources, 2006. 155(2): p. 401-414.
20. Johnson, B.A. and R.E. White, Characterization of commercially available
lithium-ion batteries. Journal of Power Sources, 1998. 70(1): p. 48-54.
21. Franco, A.A., Multiscale Modeling of Rechargeable Lithium Ion Batteries:
Concepts, Methods and Challenges. RSC Advances, 2013.
22. Arora, P., et al., Comparison between computer simulations and experimental
data for high-rate discharges of plastic lithium-ion batteries. Journal of Power
Sources, 2000. 88(2): p. 219-231.
23. Section, G.E.C.B.P., D.L. Barney, and R.L. Falzone, Nickel-cadmium battery
application engineering handbook1971: General Electric.
24. Thackeray, M.M., J.O. Thomas, and M.S. Whittingham, Science and
Applications of Mixed Conductors for Lithium Batteries. MRS Bulletin, 2000.
25(03): p. 39-46.
25. Besenhard, J.O., J. Yang, and M. Winter, Will advanced lithium-alloy anodes
have a chance in lithium-ion batteries? Journal of Power Sources, 1997. 68(1):
p. 87-90.
26. van Schalkwijk, W. and B. Scrosati, Advances in Lithium-Ion Batteries2002:
Springer.
27. Chen, J., C. Buhrmester, and J.R. Dahn, Chemical Overcharge and
Overdischarge Protection for Lithium-Ion Batteries. Electrochemical and Solid-
State Letters, 2005. 8(1): p. A59-A62.
28. Arora, P., R.E. White, and M. Doyle, Capacity Fade Mechanisms and Side
Reactions in Lithium‐Ion Batteries. Journal of The Electrochemical Society,
1998. 145(10): p. 3647-3667.
29. Lijun, G., S. Liu, and R.A. Dougal, Dynamic lithium-ion battery model for
system simulation. Components and Packaging Technologies, IEEE
Transactions on, 2002. 25(3): p. 495-505.
30. Vetter, J., et al., Ageing mechanisms in lithium-ion batteries. Journal of Power
Sources, 2005. 147(1–2): p. 269-281.
1.Introduction
22
31. Broussely, M., et al., Main aging mechanisms in Li ion batteries. Journal of
Power Sources, 2005. 146(1–2): p. 90-96.
32. Broussely, M., et al., Aging mechanism in Li ion cells and calendar life
predictions. Journal of Power Sources, 2001. 97–98(0): p. 13-21.
33. Bruce, P.G., B. Scrosati, and J.-M. Tarascon, Nanomaterials for Rechargeable
Lithium Batteries. Angewandte Chemie International Edition, 2008. 47(16): p.
2930-2946.
34. Linden, D. and T.B. Reddy, Handbook of Batteries2002: McGraw-Hill.
35. Kiehne, H.A., Battery Technology Handbook2003: Marcel Dekker.
36. Pistoia, G., Batteries for Portable Devices2005: Elsevier Science.
37. Broussely, M., et al., Lithium-ion batteries for electric vehicles: performances of
100 Ah cells. Journal of Power Sources, 1997. 68(1): p. 8-12.
38. Kennedy, B., D. Patterson, and S. Camilleri, Use of lithium-ion batteries in
electric vehicles. Journal of Power Sources, 2000. 90(2): p. 156-162.
39. Zhang, X., et al., Diagnostic Characterization of High Power Lithium-Ion
Batteries for Use in Hybrid Electric Vehicles. Journal of The Electrochemical
Society, 2001. 148(5): p. A463-A470.
40. Holdren, J.P., The Energy Innovation Imperative: Addressing Oil Dependence,
Climate Change, and Other 21st Century Energy Challenges. Innovations:
Technology, Governance, Globalization, 2006. 1(2): p. 3-23.
41. Mabro, R.E. and O.o.P.E. Countries, Oil in the 21st Century: Issues, Challenges
and Opportunities2006: Oxford University Press.
42. Cairns, E.J., & Albertus, P. , Batteries for electric and hybrid-electric vehicles.
Annual Review of Chemical and Biomolecular Engineering, 2010. 1: p. 299–
320.
43. Chu, A. and P. Braatz, Comparison of commercial supercapacitors and high-
power lithium-ion batteries for power-assist applications in hybrid electric
vehicles: I. Initial characterization. Journal of Power Sources, 2002. 112(1): p.
236-246.
44. Smith, K. and C.-Y. Wang, Power and thermal characterization of a lithium-ion
battery pack for hybrid-electric vehicles. Journal of Power Sources, 2006.
160(1): p. 662-673.
45. Burke, A.F., Batteries and Ultracapacitors for Electric, Hybrid, and Fuel Cell
Vehicles. Proceedings of the IEEE, 2007. 95(4): p. 806-820.
1.Introduction
23
46. Yuan, X., H. Liu, and J. Zhang, Lithium-ion Batteries: Advanced Materials and
Technologies2011: Taylor & Francis.
47. Yoshio, M., R.J. Brodd, and A. Kozawa, Lithium-Ion Batteries: Science and
Technologies2009: Springer-Verlag New York.
48. Aravindan, V., et al., LiMnPO4 - A next generation cathode material for
lithium-ion batteries. Journal of Materials Chemistry A, 2013. 1(11): p. 3518-
3539.
49. Hayner, C.M., X. Zhao, and H.H. Kung, Materials for Rechargeable Lithium-
Ion Batteries. Annual Review of Chemical and Biomolecular Engineering, 2012.
3(1): p. 445-471.
50. Zhang, W.-J., A review of the electrochemical performance of alloy anodes for
lithium-ion batteries. Journal of Power Sources, 2011. 196(1): p. 13-24.
51. Meyer, W.H., Polymer Electrolytes for Lithium-Ion Batteries. Advanced
Materials, 1998. 10(6): p. 439-448.
52. Song, J.Y., Y.Y. Wang, and C.C. Wan, Review of gel-type polymer electrolytes
for lithium-ion batteries. Journal of Power Sources, 1999. 77(2): p. 183-197.
53. Wang, G.X., et al., Nanosize cobalt oxides as anode materials for lithium-ion
batteries. Journal of Alloys and Compounds, 2002. 340(1–2): p. L5-L10.
54. Liaw, B.Y., et al., Modeling capacity fade in lithium-ion cells. Journal of Power
Sources, 2005. 140(1): p. 157-161.
55. Ramadesigan, V., et al., Modeling and Simulation of Lithium-Ion Batteries from
a Systems Engineering Perspective. Journal of The Electrochemical Society,
2012. 159(3): p. R31-R45.
56. Vu, A., Y. Qian, and A. Stein, Porous Electrode Materials for Lithium-Ion
Batteries – How to Prepare Them and What Makes Them Special. Advanced
Energy Materials, 2012. 2(9): p. 1056-1085.
57. Chung, G.C., et al., Effect of Surface Structure on the Irreversible Capacity of
Various Graphitic Carbon Electrodes. Journal of The Electrochemical Society,
1999. 146(5): p. 1664-1671.
58. Inagaki, M., Carbon materials Structure, texture and intercalation. Solid State
Ionics, 1996. 86–88, Part 2(0): p. 833-839.
59. Tran, T.D., et al., Commercial Carbonaceous Materials as Lithium Intercalation
Anodes. Journal of The Electrochemical Society, 1995. 142(10): p. 3297-3302.
1.Introduction
24
60. Delmas, C. and I. Saadoune, Electrochemical and physical properties of the
LixNi1−yCoyO2 phases. Solid State Ionics, 1992. 53–56, Part 1(0): p. 370-375.
61. Myung, S.-T., et al., Structural, Electrochemical, and Thermal Aspects of
Li [ ( Ni0.5Mn0.5 ) 1 − x Co x ] O2 ( 0 ≤ x ≤ 0.2 ) for High-Voltage Application
of Lithium-Ion Secondary Batteries. Journal of The Electrochemical Society,
2008. 155(5): p. A374-A383.
62. Shin, J.H., et al., Rechargeable Li/LiFePO4 cells using N-methyl-N-butyl
pyrrolidinium bis(trifluoromethane sulfonyl)imide–LiTFSI electrolyte
incorporating polymer additives. Electrochimica Acta, 2008. 54(2): p. 410-414.
63. Wang, G.X., et al. Structural and electro-chemical characteristics of Li1+x
Mn2−x O4 and LiMn2 O4 for Secondary Lithium batteries. in Battery
Conference on Applications and Advances, 1998., The Thirteenth Annual. 1998.
64. Han, D.-W., et al., Tailoring Crystal Structure and Morphology of LiFePO4/C
Cathode Materials Synthesized by Heterogeneous Growth on Nanostructured
LiFePO4 Seed Crystals. ACS Applied Materials & Interfaces, 2013. 5(4): p.
1342-1347.
65. Fergus, J.W., Recent developments in cathode materials for lithium ion
batteries. Journal of Power Sources, 2010. 195(4): p. 939-954.
66. de las Casas, C. and W. Li, A review of application of carbon nanotubes for
lithium ion battery anode material. Journal of Power Sources, 2012. 208(0): p.
74-85.
67. Iijima, S., Helical microtubules of graphitic carbon. Nature, 1991. 354(6348): p.
56-58.
68. Arora, P. and Z. Zhang, Battery Separators. Chemical Reviews, 2004. 104(10):
p. 4419-4462.
69. A, M.S., Review on gel polymer electrolytes for lithium batteries. European
Polymer Journal, 2006. 42(1): p. 21-42.
70. Millet, P., F. Andolfatto, and R. Durand, Preparation of solid polymer
electrolyte composites: investigation of the precipitation process. Journal of
Applied Electrochemistry, 1995. 25(3): p. 233-239.
71. Xu, K., Nonaqueous Liquid Electrolytes for Lithium-Based Rechargeable
Batteries. Chemical Reviews, 2004. 104(10): p. 4303-4418.
72. Saunier, J., et al., Plasticized microporous poly(vinylidene fluoride) separators
for lithium-ion batteries. I. Swelling behavior of dense membranes with respect
1.Introduction
25
to a liquid electrolyte—Characterization of the swelling equilibrium. Journal of
Polymer Science Part B: Polymer Physics, 2004. 42(3): p. 532-543.
73. Saunier, J., et al., Plasticized microporous poly(vinylidene fluoride) separators
for lithium-ion batteries. III. Gel properties and irreversible modifications of
poly(vinylidene fluoride) membranes under swelling in liquid electrolytes.
Journal of Polymer Science Part B: Polymer Physics, 2004. 42(12): p. 2308-
2317.
74. Li, W. and J.R. Dahn, Lithium‐Ion Cells with Aqueous Electrolytes. Journal of
The Electrochemical Society, 1995. 142(6): p. 1742-1746.
75. Barthel, J., R. Wachter, and H.J. Gores, Temperature Dependence of
Conductance of Electrolytes in Nonaqueous Solutions, in Modern Aspects of
Electrochemistry, B.E. Conway and J.O.M. Bockris, Editors. 1979, Springer US.
p. 1-79.
76. Xiao, Q., et al., A novel sandwiched membrane as polymer electrolyte for
application in lithium-ion battery. Journal of Membrane Science, 2009. 326(2):
p. 260-264.
77. Wachtler, M., et al., A study on PVdF-based SiO2-containing composite gel-type
polymer electrolytes for lithium batteries. Electrochimica Acta, 2004. 50(2–3):
p. 357-361.
78. Webber, A., Conductivity and Viscosity of Solutions of LiCF3 SO 3,
Li ( CF 3 SO 2 ) 2 N , and Their Mixtures. Journal of The Electrochemical
Society, 1991. 138(9): p. 2586-2590.
79. Muniyandi, N., et al., Optimisation of PVdF-based polymer electrolytes. Journal
of Power Sources, 2001. 96(1): p. 14-19.
80. Yang, H., G.V. Zhuang, and P.N. Ross Jr, Thermal stability of LiPF6 salt and
Li-ion battery electrolytes containing LiPF6. Journal of Power Sources, 2006.
161(1): p. 573-579.
81. Angulakshmi, N., et al., Electrochemical and mechanical properties of
nanochitin-incorporated PVDF-HFP-based polymer electrolytes for lithium
batteries. Ionics, 2011. 17(5): p. 407-414.
82. Shimizu, H., et al., Sol-gel transitions of poly(vinylidene fluoride) in organic
solvents containing LiBF4. Polym J, 2011. 43(6): p. 540-544.
83. Verdolotti, L., et al., Effects of the addition of LiCl, LiClO4, and LiCF3SO3
salts on the chemical structure, density, electrical, and mechanical properties of
1.Introduction
26
rigid polyurethane foam composite. Polymer Engineering & Science, 2011.
51(6): p. 1137-1144.
84. Park, J.K., Principles and Applications of Lithium Secondary Batteries2012:
Wiley.
85. Chung, Y.S., S.H. Yoo, and C.K. Kim, Enhancement of Meltdown Temperature
of the Polyethylene Lithium-Ion Battery Separator via Surface Coating with
Polymers Having High Thermal Resistance. Industrial & Engineering Chemistry
Research, 2009. 48(9): p. 4346-4351.
86. Venugopal, G., et al., Characterization of microporous separators for lithium-
ion batteries. Journal of Power Sources, 1999. 77(1): p. 34-41.
87. Kang, Y., et al., Photocured PEO-based solid polymer electrolyte and its
application to lithium–polymer batteries. Journal of Power Sources, 2001. 92(1–
2): p. 255-259.
88. Choi, B.K., K.H. Shin, and Y.W. Kim, Lithium ion conduction in PEO–salt
electrolytes gelled with PAN. Solid State Ionics, 1998. 113–115(0): p. 123-127.
89. Cho, T.-H., et al., Battery performances and thermal stability of
polyacrylonitrile nano-fiber-based nonwoven separators for Li-ion battery.
Journal of Power Sources, 2008. 181(1): p. 155-160.
90. Djian, D., et al., Macroporous poly(vinylidene fluoride) membrane as a
separator for lithium-ion batteries with high charge rate capacity. Journal of
Power Sources, 2009. 187(2): p. 575-580.
91. Gao, K., et al., Crystal structures of electrospun PVDF membranes and its
separator application for rechargeable lithium metal cells. Materials Science
and Engineering: B, 2006. 131(1–3): p. 100-105.
92. Shen, Y.J., M.J. Reddy, and P.P. Chu, Porous PVDF with LiClO4 complex as
‘solid’ and ‘wet’ polymer electrolyte. Solid State Ionics, 2004. 175(1–4): p. 747-
750.
93. Costa, C.M., et al., Effect of degree of porosity on the properties of
poly(vinylidene fluoride–trifluorethylene) for Li-ion battery separators. Journal
of Membrane Science, 2012. 407–408(0): p. 193-201.
94. Miao, R., et al., PVDF-HFP-based porous polymer electrolyte membranes for
lithium-ion batteries. Journal of Power Sources, 2008. 184(2): p. 420-426.
95. Costa, C.M., et al., Evaluation of the main processing parameters influencing
the performance of poly(vinylidene fluoride–trifluoroethylene) lithium-ion
1.Introduction
27
battery separators. Journal of Solid State Electrochemistry, 2013. 17(3): p. 861-
870.
96. Srun Jung, et al., Fillers for Solid-State Polymer Electrolytes: Highlight. Bull.
Korean Chem. Soc, 2009. 30(10).
97. Cai, L. and R.E. White, Mathematical modeling of a lithium ion battery with
thermal effects in COMSOL Inc. Multiphysics (MP) software. Journal of Power
Sources, 2011. 196(14): p. 5985-5989.
98. Sikha, G., R.E. White, and B.N. Popov, A Mathematical Model for a Lithium-
Ion Battery/Electrochemical Capacitor Hybrid System. Journal of The
Electrochemical Society, 2005. 152(8): p. A1682-A1693.
99. Chirkov, Y.G., V.I. Rostokin, and A.M. Skundin, Computer simulation of
operation of lithium-ion battery: Galvanostatics, central problem of theory,
calculation of characteristics of thin active layers with low diffusion coefficients.
Russian Journal of Electrochemistry, 2011. 47(11): p. 1239-1249.
100. Subramanian, V.R., V. Boovaragavan, and V.D. Diwakar, Toward Real-Time
Simulation of Physics Based Lithium-Ion Battery Models. Electrochemical and
Solid-State Letters, 2007. 10(11): p. A255-A260.
101. Ramadesigan, V., et al., Efficient Reformulation of Solid-Phase Diffusion in
Physics-Based Lithium-Ion Battery Models. Journal of The Electrochemical
Society, 2010. 157(7): p. A854-A860.
102. Subramanian, V.R., et al., Mathematical Model Reformulation for Lithium-Ion
Battery Simulations: Galvanostatic Boundary Conditions. Journal of The
Electrochemical Society, 2009. 156(4): p. A260-A271.
103. Gomadam, P.M., et al., Mathematical modeling of lithium-ion and nickel battery
systems. Journal of Power Sources, 2002. 110(2): p. 267-284.
104. Garcı́a, R.E., et al., Microstructural Modeling and Design of Rechargeable
Lithium-Ion Batteries. Journal of The Electrochemical Society, 2005. 152(1): p.
A255-A263.
105. Sikha, G., B.N. Popov, and R.E. White, Effect of Porosity on the Capacity Fade
of a Lithium-Ion Battery: Theory. Journal of The Electrochemical Society, 2004.
151(7): p. A1104-A1114.
106. Ferguson, T.R. and M.Z. Bazant, Nonequilibrium Thermodynamics of Porous
Electrodes. Journal of The Electrochemical Society, 2012. 159(12): p. A1967-
A1985.
1.Introduction
28
107. Renganathan, S., et al., Theoretical Analysis of Stresses in a Lithium Ion Cell.
Journal of The Electrochemical Society, 2010. 157(2): p. A155-A163.
108. Dao, T.-S., C.P. Vyasarayani, and J. McPhee, Simplification and order reduction
of lithium-ion battery model based on porous-electrode theory. Journal of Power
Sources, 2012. 198(0): p. 329-337.
109. Lee, J.-W., Y.K. Anguchamy, and B.N. Popov, Simulation of charge–discharge
cycling of lithium-ion batteries under low-earth-orbit conditions. Journal of
Power Sources, 2006. 162(2): p. 1395-1400.
110. Martinez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
2. State of the art
29
2. State of the art on microscopic theoretical models and
simulations of lithium-ion rechargeable batteries
This chapter describes the state of art on the theoretical models for the simulation of
the performance of lithium ion batteries. The main theoretical studies that describe the
operation and performance of a battery are presented. Finally, the influence of the most
relevant parameters of the models, such as boundary conditions, geometry and material
characteristics are discussed.
This chapter is based on the following publication:
“Lithium ion rechargeable batteries: State of the art and future needs of
microscopic theoretical models and simulations”, D. Miranda, C.M. Costa, S.
Lanceros-Mendez, Journal of Electroanalytical Chemistry 739 (2015) 97-110.
2. State of the art
31
2.1 Microscopic modelling of lithium ion batteries
With the appearance of lithium batteries, several theoretical studies and models
have been performed for understanding their main processes and for improving their
performance. The developed models include parameters for the understanding of
materials and microstructure of the electrodes, the most suitable organic solvents for
electrolytes, geometry, dimensions of the different components of the battery and the
materials and microstructure of the separator, among other variables [1–12].
Simulations and modeling have been performed through different programming
languages, including C++ [13], MatLab [14], Simulink [15], Fluent [16], Battery Design
Studio [17] and COMSOL Multiphysics [18], among others.
The microscopic models for the operation of lithium-ion batteries are based on the
mathematical expressions of the fundamental physical and chemical processes
associated to the electrochemical phenomena, ionic diffusion and mass transport.
However, the models are simplified according to the boundary conditions selected as a
function of the main goals of each study. Some models introduce also thermal
conditions. The research in the development of models for lithium-ion batteries
introduced important parameters in battery performance such as the parameter of
porosity for electrodes and separators.
The majority of the theoretical models using the electrodes LixC6–LiyMn2O4 are
based on the Doyle/Fuller/Newman model [19,20] with specific boundary conditions
[21] in order to describe the three components of battery.
With the evolution of the complexity and accuracy of the models, higher processing
time was required and reformulation of the mathematical models had to be performed in
order to improve computational efficiency [5].
In this way, a simplified model for lithium-ion batteries based on the porous-
electrode theory was presented [11], Figure 2.1. The model incorporates the
concentrated solution theory, the porous electrode theory, and the variations in
electronic/ionic conductivities and diffusivities, the simplification being based in
exploiting the nature of the model and the structure of the governing equations.
2. State of the art
32
Figure 2.1 - 1C discharge voltage curve comparison between the rigorous model and
the simplified model at different number of terms or node points through the Galerkin’s
approximation. Figure from [11].
The simplification of the model has been achieved through the Galerkin’s
approximation, which allows converting a continuous operator problem into a discrete
problem, allowing to reduce computational time significantly while still retaining the
accuracy compared to the full-order rigorous model.
A major difficulty to simulate lithium-ion batteries is the need to account for
diffusion in the solid phase (active material) taking into account the spherical
coordinates (dimension r). This fact increases the complexity of the models developed
for lithium-ion batteries, as well as the computation time. In this context, a
computationally efficient representation for solid phase diffusion was presented in [4]
using an eigenfunction based Galerkin method and a mixed order finite difference
method for approximating/representing solid-phase concentration variations within the
active materials of the porous electrodes for a pseudo-two dimensional model for
lithium ion batteries.
The complexity of the battery systems affects the speed and accuracy of the
different numerical methods including operating and boundary conditions at the
microscale.
2. State of the art
33
2.2 Simulation of the components of the battery: electrodes and
separator/electrolyte
In this section, the main results of the theoretical simulation developed for the
different components of the lithium ion battery (electrodes and separator/electrolyte)
will be presented. The models account for the study of undesirable phenomena in the
battery performance (e.g. deposition of lithium ions at the cathode), the influence of the
dimensions of the electrodes, the porosity and the particle size of the active material, as
well as for a better under- standing of ionic conduction phenomena in lithium ion
batteries. Typically, the theoretical models of the lithium-ion battery are 1-D and 2-D,
being also developed models for spirally wound cells [22] in 3-dimensions [23].
2.2.1 Electrodes
In the study of lithium ion batteries, it was verified the importance given to
phenomena occurring at the interface between electrodes and electrolytes (margins or
edges). These effects were accounted for in the model of Kennell & Evitts [24] which
focused on the prediction of the effects associated to electrode length and extent of the
cathode and electrolyte in lithium ion batteries (Figure 2.2).
Figure 2.2 - Cell configuration (not to scale). The x-dimension corresponds to the
length of the cell and the y-dimension corresponds to the height of the cell. Figure
adapted from [24].
2. State of the art
34
Lithium ions are produced and consumed at the electrode/electrolyte interfaces
when the electrode comes into contact with the electrolyte, for example, when the
electrolyte overflows the edges of the electrodes. At the ends of flooded electrodes
(edges) the edge geometry can cause multidimensional effects, such as concentration
gradients in the electrolyte and the electrodes, and also electrical potential gradients in
the electrolyte. In [24], the authors explored the effects of the edges of the electrodes of
the battery during charging and the effect of the gradient of the stoichiometric
coefficient inside the electrode. It was shown that increasing effective conductivity
relative to the electrolyte which extends beyond the edges of the electrodes does not
have a significant effect on the rates of the anodic and cathodic reactions occurring at
the edge regions of the electrodes. Furthermore, it is predicted that whereas lithium
concentration gradients within the cathode have an impact on reaction rates of the
cathode, lithium concentration gradients inside the anode have no significant impact on
the rates of the anodic reactions during the early charge cell. It was verified that the
rates of the anodic reactions are significantly affected by the surface area of the anode
that is in contact with the electrolyte and not by the concentration gradient of lithium at
the anode. It was also concluded that during the final stages of battery charge, the
concentration gradients within the cathode (for equal lengths of electrodes) are more
likely and may lead to deposition of lithium on the edge region of the cathode. In this
study, simulations were performed for the case in which the tip of the cathode was
extended beyond the edge of the anode to reduce the possibility of deposition of lithium
at the edge region of the cathode. The simulations indicate that stoichiometric lithium in
an extended edge of the cathode would be of little value, however, this extension may
cause a high electric potential drop along the length of electrolyte during the initial
battery charge. It was observed that a decreasing gradient equilibrium potential during
charging of the battery causes a reduction in the rate of cathodic reactions which occurs
along the extended cathode. This reduction in cathodic reactions along the extended
region of the cathode reduces the risk for deposition of lithium on the cathode edge.
It is thus important to avoid the negative consequences for the performance of the
cell that may arise due to concentration gradients associated with edges (interfaces) of
the electrodes flooded by the electrolyte. These consequences include an increased risk
of deposition of lithium on the cathode region. Therefore, the cathode extension beyond
the edge of the anode can reduce the probability of deposition of lithium on the cathode
2. State of the art
35
edge region. On the other hand, this may result in other problems, such as high drop in
electrical potential along the length of the electrolyte in parallel with the electrodes and
associated with the extended edge of the cathode [25,26].
West et al. [27] developed a one-dimensional model using porous electrodes and a
liquid electrolyte, demonstrating that depletion in the electrolyte was the main factor
that limits the discharge capacity of the battery. This depletion is a consequence of the
mobility of the non-inserted ions, so the performance of this type of electrode is
optimized by the choice of electrolyte through of the number of transport as close to
unity as possible for the inserted ion.
Doyle et al. [19] presented a one dimensional model for a lithium ion battery and
verified that the concentration of lithium decreased on the cathode material, illustrating
the necessity of high concentrations of lithium. This model was developed in [28]
considering a porous anode rather than a lithium foil anode. Transport in the electrolyte
is described within the scope of the concentrated solution theory in the LixC6/LiyMn2O4
system with 1 M LiClO4 in PC. Further, a two-dimensional model was also developed
for the investigation of deposition of lithium [29] assuming:
– Concentration of electrolyte and conductivity are constant and uniform.
– Uniform concentration at the anode (the same concentration along the anode).
– Application of the linearization model of the Tafel kinetics.
– Solid film electrodes.
– Electroneutrality of the electrolyte.
It was shown that extending the cathode 0.4 mm is enough to prevent the occurrence
of deposition of lithium.
Eberman et al. [30] used a two dimensional model based on the theory of
concentrated solution for modeling the effects of decreased dimensions of a cathode in
order to perform an analysis of various parameters on the risk of deposition of lithium.
It was found that the three most important factors that affect the deposition are the open
circuit potential, the rate of decreasing of the dimensions of the cathode and the
charging rate.
A further two dimensional model for the study of the effects of concentration,
distribution of current and electric field versus time profile in a lithium ion batteries
[24] demonstrated that it is possible to predict not only the deposition of lithium on the
cathode edge during higher charge times, but also the high electric gradients which were
2. State of the art
36
observed experimentally in [25,26] along the electrolyte during the initial charging.
The influence of the variation of the electrode width and porosity of the electrodes
in battery performance was also studied [21] leading to the conclusion that the width of
the electrodes deter- mines two main factors in the function of the battery: the quantity
of active material and the resistance to mass transport. The width of both electrodes was
varied uniformly in a range of 80–120% of the baseline. It was found a slight increase in
battery capacity when the width of the electrode increases. Further, the porosity of the
electrodes affects the effective conductivity and the resistance to mass transfer. The
variation of the porosity was performed in the same range used for the study of the
width of the electrodes and it was verified that there is no linear relationship but the
parameters can be optimized.
For plastic lithium-ion batteries it was developed a simulation model taking account
the thickness value of the electrodes, active material loadings and initial salt
concentrations with the objective of better understanding the transport processes of the
plasticized polymer electrolyte system [31] in Bellcore PLION cells. The results
obtained in the simulation were compared with experimental data as shown in Figure
2.3.
Figure 2.3 - Experimental and simulated discharge curves for PLION cells at low
rates. The C rates for thin, medium and thick cells are 2.312, 2.906, and 3.229 mA/cm2,
respectively. The dots represent the experimental data and the solid lines correspond to
the simulation results. Figure from [31].
2. State of the art
37
It is observed a good agreement between simulation and experimental data due to
the use of a contact resistance at the interface between the current collector and the
electrode, this being an adjustable parameter for different batteries. The diffusion
coefficient of the salt at high discharge rates was also reduced to approximate the results
of simulation with the experimental ones.
Regarding the dimensions of the electrodes (fine, medium and thick batteries) the
diffusion limitations are most significant for thicker than fine and medium batteries and
the limitations of diffusion in the solution phase are the main limiting factor for proper
battery performance at high rate discharge [31].
Battery systems with lithium and nickel [6] have been simulated to account for the
behavior of particles in a single electrode, individual cells and complete batteries
(complete set of cells) based on varying operation conditions such as constant current
discharge, pulse discharge, cyclic voltammetry and impedance. A review of the
theoretical models for nickel, simulating the performance of complete cells, including
the behavior of the active material (nickel hydroxide) was presented. It was concluded
that the diffusion coefficient depends on the size of the cells as shown in Figure 2.4.
Figure 2.4 - Solution phase diffusion coefficient as a function of discharge rate used to
fit experimental data for three different cells. 1C corresponds to 1.156, 1.937 and 2.691
A/m2 for thin, medium and thick cells, respectively. Figure from [6].
In the study (LiyC6/LixMn2O4) [7], the properties of recharge- able lithium-ion
batteries was calculated considering the electro- chemical properties of the materials
and focusing on the influence of their microstructure. The main conclusion is that the
2. State of the art
38
battery performance can be improved by controlling the transport paths to the back of
the porous positive electrode, maximizing the surface area for intercalating lithium ions,
and carefully controlling the distribution and particle size of the active material.
The model developed for the calculation of the effect of porosity on the capacity
fade of a lithium-ion battery [8] includes the changes in the porosity of the material due
to the reversible intercalation processes and irreversible parasitic reactions. Thus, a
general method for the capacity fade prediction of a lithium ion battery system was
presented. The variation in porosity due to the side reaction products formed during
cycling causes the discharge voltage plateau to drop with cycling.
With respect to the theoretical analysis of stresses in a lithium ion cell [10], the
mathematical model is developed to simulate the generation of mechanical stress during
the discharge process in a dual porous insertion electrode cell sandwich comprised of
lithium cobalt oxide and carbon. This model shows that the accumulation of stress
within intercalation electrodes leads to changes in the lattice volume due to the
intercalation and phase transformation during charge/discharge. The model provides the
main parameters influencing the magnitude of the battery generation of stress, such as
thickness, porosity and particle size of the electrodes. The developed model is used to
understand the mechanical degradation of a porous electrode during the process of
insertion/ extraction of lithium ions.
Other studies are based on the effort to gain a better under- standing of conduction
phenomena of the lithium ion [32] in order to allow innovative technologies and a
comprehensive understanding of the phenomena of conduction in all components of a
lithium-ion battery incorporating theoretical analyses of the fundamentals of electrical
and ionic conduction at the cathode, anode and electrolyte. A review of the relationship
between electrical and ionic conduction of three cathode materials: LiCoO2, LiMn2O4,
LiFePO4, is presented in [32], discussing the phase shift in graphite anodes and how
they relate to diffusivity and conductivity. The phenomenon of electrical and ionic
conduction has been one of the main objectives of the study for the development of
models of lithium-ion batteries. The review work presented in [32] refers to various
aspects of this problem that have been dis- cussed for each of the main components of a
lithium-ion battery (anode, cathode and electrolyte) as previously stated.
Efforts to optimize the electrical and ionic conductivity and the cathode have
focused largely on doping methods (liquid–solid method, spray drying method, etc.) to
2. State of the art
39
improve the electrical conductivity and ionic conductivity. Viable methods for
improving the electrical conductivity are based on covering the cathode surface using a
conductive material and by using micro and nanoparticles [32] as shown in Figure 2.5
for LiFePO4 active material.
Figure 2.5 - Conduction phenomena in the LiFePO4 cathode during battery charging.
Figure from [32].
Mathematical developments suggest that fibrous architectures, such as carbon–
silicon-nanocomposites, show better results with respect to improving ionic
conductivity [33].
With respect to the anodes, importance is attributed to the intercalation process.
The most commonly used materials for the anode are carbonaceous materials including
graphites (natural graphite and Highly Ordered Pyrolytic Graphite (HOPG)), modified
graphites (MesoCarbon MicroBeads (MCMB), carbon fiber, metal deposited carbon
fiber) and non-graphitic carbons. The diffusivity of lithium ion in graphite is
complicated by the constant change in the phase intercalation compound Li–graphite,
which can cause disorder in the original structure [32].
It is observed that with increasing degree of intercalation the diffusivity becomes
smaller. For this reason, the diffusivity must be understood as a function of electrode
voltage or intercalation. Further, the optimization between the crystalline and
amorphous phases is an important strategy for improving conductivity in carbon
electrodes.
Computer simulation models [2] are used for studying the operation of a lithium-
ion battery discharge galvanostatic mode based on the central problem and calculation
of the characteristics of thin active layers with low diffusion coefficients. The authors
2. State of the art
40
showed the mathematical model of the processes occurring in the active layers of the
electrodes. The central problem of the theory of lithium ion batteries is the possibility of
analyzing two processes in space and time: the recovery or filling of active substance
(intercalating agent) grains with lithium atoms and redistribution of electrode potentials
over the active layer width, which related to ohmic limitations. The authors report that
the diffusion coefficient of lithium atoms in the grain intercalating agent is important. In
the electrodes with high diffusion coefficients, the size of the intercalating agent grain is
limited, whereas in the electrodes with low diffusion coefficient, there are no
restrictions on the grain size.
In this study, the advantages and disadvantages of the electrodes in relation to the
high and low diffusion coefficients are reported. The calculation of these parameters of
the electrolyte is achieved for active layers with low diffusion coefficients such as ≤10-
13 cm2/s. The thickness of the active layer, the time of full dis- charge, electrical
capacitance and specific potential within the interface layer/interactive electrodes are
determined where the grain size is commonly limited to approximately 10m.
The importance of the electrode design, i.e., its performance through of the
optimization of parameters (the weight fraction of electronic particle additives,
electrode thickness and electrode density or porosity) was shown in [34]. Numerical
model simulation also proved that ion transport in the electrolyte phase becomes more
difficult in dense electrodes and that high electrode compression to obtain high energy
density may cause severe transport loss. However, the discharge current will decrease
with increasing grain size.
Chen et al. [35], showed that design of cathode electrodes for high specific energy
also creates higher operation and specific power. It is observed that for improving the
performance of the cathode, the most important issue is to properly consider the cathode
thickness and volume fraction of active material with respect to ion transportation,
cathode capacity and mass balance effect of active material. The addition of additives
(for example, carbon back) improves the specific energy through optimization of the
cathode composition and cathode design but penalizes both volumetric and gravimetric
properties of the cathode.
The prediction of the impedance response of a dual insertion electrode cell separated
by an ionically conduction membrane was presented in [36]. The used expressions take
into account the reaction kinetics and double-layer adsorption processes at the
2. State of the art
41
electrode–electrolyte interface, transport of electroactive species in the electrolyte phase
and solid phase of the electrodes. The prediction of the impedance response was
obtained through the analytical expression development of a lithium-ion cell consisting
on a porous LiCoO2 cathode and mesocarbon microbead anode [36].
The lithium-ion concentration profile simulation in the cathode for a half-cell was
studied in [37]. The model used in this work describes the discharge behavior of a
rechargeable cell based on the simulated concentration profile. The cathode material
used of this work was LiMn2O4.
In secondary batteries, the battery design was optimized through the efficient design
of porous electrodes using a physics based porous electrode theory model with
increased computational efficiency [38]. This model optimizes the discharge capacity
given size constraints, rather than time constraints and minimizing the temperature
gradient across a cell for sage operation and prevention of thermal runaway.
Cooper et al., quantified the effect of tortuosity of the porous electrode on the
diffusion through the material. No correlation was observed between the measured
tortuosities and the ones determined by Bruggeman equation [39] in which an isotropic
and homogeneous material is considered.
These simulations demonstrate thus that tortuosity is not a simple scalar quantity,
but instead both geometrical and transport tortuosities show a marked dependence on
direction, i.e., a vectorial representation of tortuosity should be developed [39].
The capacity of lithium-ion batteries has been improved by adding conducting
species in the battery materials, more specifically the cathode [35]. The addition of
conducting species shall not limit the transport and performance at high discharge rates.
This work [35] developed a technique to optimize the cathode with respect to ionic and
electrical conductivity and specific energy. Figure 2.6 illustrates the composition of the
simulated structure of the complementary solid phase and electrolyte phase obtained by
finite element conduction model.
2. State of the art
42
Figure 2.6 - Illustration of the composition of the cathode electrode: complementary
solid phase and electrolyte phase [35].
It is shown in this way the importance of the design of the cathode and how to
optimize the composition of the cathode with additives for improved specific energy.
The influence of different electrode (LiFePO4) parameters on the performance of
the battery, including conflicting effects of the conductor ratio (the weight fraction of
electronic particle additives), electrode thickness and density (porosity), were addressed
on the basis of experimental results and simulations [34]. In the context of the
simulation model, it was concluded that the transport of ions decreases for thicker
electrodes. Although the compression of the electrodes increases the energy density, it
can cause a decrease in ion transport by reducing the diffusion and ionic conductivity
from the electrolyte phase to the electrode. Further, at high discharge rates, very thick or
very dense electrodes show a significant loss of tension due to a slowdown of the
transport of ions in the liquid phase (so-called limited transport). Increased thickness
and density of the electrode above a certain critical values lead to a small increase in the
discharge capacity of the cell [34].
The influence of the microstructural morphology of the electrode (LiCoO2,
LiFePO4 and LiMn2O4) in the performance of the battery is analyzed in [40]. An
analytical approach is presented that properly reproduces the experimental results
obtained after measuring the resistance of the electrode, capturing the most important
effects of the microstructure of the electrode in battery performance. For LiCoO2 and
LiFePO4 the relevance of solid surface characteristics and microstructure are significant
due to losses in the electrical charge transport efficiency, including reduced charge
2. State of the art
43
transfer kinetics [40].
The effect of tortuosity anisotropy in lithium-ion battery electrodes was shown for
LiNi1/3Mn1/3Co1/3O2 and LiCoO2 [41]. For these active materials, Bruggeman exponents
are estimated to be 0.66 and 1.94 for LiNi1/3Mn1/3Co1/3O2 and LiCoO2, respectively.
These Bruggeman exponents are in agreement with those calculated through the
numerical diffusion simulations performed directly on the tomography data [41].
Independently of the active material, alignment of the particles parallel to the current
collector during electrode manufacturing affects the tortuosity and porosity value of the
electrodes.
Mathematical models for lithium-ion cells with blended-electrodes were also
developed [42]. These dynamic models allow simulations under various operating
conditions such as C-rate and temperature by solving physico-chemical governing
equations. The results of the models show good agreement with the experiment data,
confirming that the present model is useful for evaluating possible active material
compositions [42].
2.2.2 Separator and electrolyte
The key issues for conduction in organic liquid, solid state electrolytes and ionic
liquids are summarized in [32], together with the ionic conductivity for various
electrolytes (organic solvents, ionic liquids and electrolytes in solid state) indicating that
LiPF6 in 1M EC/DMC shows high ionic conductivity (10.7 mS/cm), rapid solvation and
good interface between electrodes but is sensitive to ambient moisture and solvents.
The study of the performance of lithium-ion batteries by varying initial
concentration of salt in the electrolyte [21] shows that the concentration of lithium ions
in the electrolyte influences the conductivity () in a non-linear way. A battery with an
initial concentration in 2000 mol/m3 gives rise to a low battery capacity ~0.6 mAh/m2.
Above this concentration, the improvements on the capacity of the cell are smaller
up to 1.8 mAh/m2, leading to the conclusion that this parameter can be optimized
contributing also to a decrease of the battery manufacturing cost.
In relation to battery separators (single polymer, composites and polymer blends) it
is verified that the ionic conductivity depends not only on the characteristics of the
electrolyte solution but also on the properties of the membrane (in particular porosity
2. State of the art
44
and pore size) [43] as also reported in [44]. Typically, the ionic conductivity of the
separator is described through of the Bruggeman equation. Theoretical and experimental
evidence show that a Bruggeman exponent of 1.5 is often not valid for real electrodes or
separator materials [44]. It was observed that only idealized morphologies, based on
spherical or ellipsoids give rise to a Bruggeman law with an exponent of about 1.3.
Polymer membranes with different morphologies or composite materials increase the
tortuous path for ionic conductivity and result either in a significant increase of the
exponent or in a complete deviation from the power law. It is found that the
MacMullin number increases with increasing anisotropy, i.e. approximately linear
function of 1/ [44]. The diffusion limitations in thick cells has been also reported [31].
Rate-dependent salt diffusion coefficients are probably an artifact of tortuous and
inhomogeneous paths for salt diffusion inside the electrode/gelled polymer regions and
reflect the inadequacy of the present simplified treatment of salt transport based on a
binary electrolyte.
In order to understand the effect of electrolyte deterioration in the performance of
Li–air batteries, a micro–macro model was constructed that includes the homogeneous
phenomenon associated with the formation of Li2CO3 that occurs by degradation of the
electrolyte during battery cycles, as shown in Figure 2.7 [45].
Figure 2.7 - Schematic computation domain of a Li–air battery during discharge
operation. The inset demonstrates the discharge products formation of Li2O2 and
Li2CO3 covering the porous carbon surface. Figure from [45].
2. State of the art
45
The deterioration of the cycle performance is measured in terms of retention of
discharge capacity, the model including the irreversible effect of the Li2CO3 in the
discharge. A good relationship between the simulation model and experimental data
was obtained, the results indicating a gradual decrease for retention of discharge
capacity with increasing number of cycles due to the effect of irreversible formation of
the Li2CO3 discharge product [45].
Due of the advances/improvements in battery separators, morphology parameters
such as porosity, pore size, tortuosity, MacMullin number and polymer density have to
be included in the computer simulation models in order to properly design and optimize
battery performance.
The knowledge and correlation between ionic conductivity, porosity, pore size,
mechanical and thermal properties are essential to achieve adequate battery separators
with high performance for lithium ion batteries. It is a new field for computer
simulation that can certainly provide new hints on battery materials optimization, in
particular with respect to future trends in which conventional electrolytes can be
replaced by gel electrolytes, ionic liquids and solid systems.
2.3 Thermal behavior simulation
In this section models for the influence of the thermal behavior in lithium ion
battery performance will be presented as well as other relevant studies for the
development and evaluation of predictive models for efficient battery performance.
A two-dimensional model for the thermal effect on the performance of lithium-ion
battery [46] was developed using a binary electrolyte and thermal conditions ranging
from adiabatic to isothermal. For adiabatic conditions, Figure 2.5, the temperature of the
battery increasing rapidly during the discharge at 1C, resulting in a higher diffusion
coefficient value for the binary electrolyte, thereby reducing the limitations of diffusion.
This fact can be verified by comparing the profile of the concentration of the electrolyte
at different cooling conditions. It was found that the concentration profile under the
adiabatic condition presents a smaller variation along the length of the battery, unlike
what happens in the case isothermal conditions. This observation indicates better
diffusion properties of the electrolyte under the adiabatic condition in relation to
isothermal condition. Figure 2.8 shows the temperature on the cell surface during 1C
2. State of the art
46
discharge process under different cooling conditions and Figure 2.9 shows the cell
voltage for 1C discharge process under different cooling conditions [46].
Figure 2.8 - Temperature on the cell surface during 1C discharge process under
different cooling conditions. Figure from [46].
Figure 2.9 - Cell voltage for 1C discharge process under different cooling conditions.
Figure from [46].
On the other hand, although good thermal insulation improves the discharge
capacity, the high temperature that the battery reaches causes an increase in the risk of
degradation of the battery.
The analysis of the electrochemical and thermal behavior of lithium ion batteries
2. State of the art
47
[47] through a model based on two-dimensional thermal-electrochemical principles
incorporating reversible, irreversible and ohmic heats in the solid and solution phases
has been performed. The temperature dependence of the various trans- port, kinetic, and
mass transfer parameters based on Arrhenius expressions are obtained.
The model incorporates experimental data on the entropic contribution for the
manganese oxide spinel and carbon electrodes with the objective of evaluating the
importance of this term in the overall heat generation.
The simulations were used to estimate the thermal and electric energy and the
active material at various rates with the objective of understanding the effect of
temperature on electrochemistry.
Simulations were performed at different rates to evaluate the importance of the
different contributions to the total heat generated in the battery. The reversible heat was
found to be important in all rates, contributing both to the final temperature of the
battery at all rates and to the evolution of the temperature during discharge.
The non-uniform reaction distribution in the porous electrode was significant at
higher rates, which in turn introduces error in estimating the heat generation based on
the average cell voltage and open-circuit voltage [47].
Predictive models for commercial lithium-ion batteries have been also performed
with the objective of evaluating the efficiency of the developed models. The study [48]
compares battery performance simulations from a commercial lithium-ion battery
modeling software package against manufacturer performance specifications and
laboratory tests to assess model validity. The authors used the Battery Design Studio ®
(BDS) software to create a mathematical model of each battery. The authors concluded
that BDS can provide sufficient accuracy in discharge performance simulations for
many applications.
An analytical model for the prediction of the remaining battery capacity of lithium-
ion batteries was presented in [49]. The model allows to predict the residual energy of
the battery source that powers a portable electronic device based on a design and
management policy for the dynamic energy efficient device. The precision of the model
was validated by comparison with simulation results DUALFOIL [50], with low
discrepancy (maximum 5%) between the predicted and simulated results.
Other models coupling thermal and electrochemical responses are developed to
predict the performance of lithium-ion batteries when those are subjected to different
2. State of the art
48
temperatures during the operation of the battery [51]. The models are in agreement with
the experimental results obtained for constant and pulsing charging and discharging
conditions characteristic of hybrid electric vehicles (HEV). This model opens the
possibility to predict and prevent situations of deposition of lithium resulting in the loss
of capacity of lithium ions battery in vehicles, allowing the study of the degradation
process and the life cycle of the batteries.
Other studies show the development of a thermal model applied to lithium ion
batteries. The models include equations related to the diffusion coefficient and reaction
rate coefficient of the electrodes as a function of temperature. These equations also
include the activation energy for diffusion and the activation energy for reaction of
electrodes, respectively Ead and Eak. The values of both activation energies depend on
the active material. Further, the ionic conductivity and diffusion coefficient of the salt in
the electrolyte as a function of temperature are also shown [52,53].
2.4 Conclusions
This chapter summarizes the main results of the theoretical models evaluating the
contributions of each of the components of a battery, anode, cathode and separator, the
performance of a lithium- ion battery. The main materials are described as well as the
main mathematical framework of the models.
Parameters affecting separator performance such as degree of porosity, pore size
and tortuosity, among others, have not been taken into account, the separator being
considered, in most studies, as a continuous medium with porosity and described by the
Bruggeman equation.
With respect to the electrodes (anode and cathode), simulations have taken into
account porosity, with no emphasis in pore size, which is a relevant parameter. The
models developed for electrodes take into account the radius of the particles of the
active material and their influence in the insertion/extraction of lithium ions to/from the
electrodes, the nature of the composite used in electrodes and their electrical
characteristics, the mechanical stability and degree of crystallinity. Theoretical models
were developed to describe the operation of a lithium-ion battery, focusing mainly on
the variation of the boundary conditions. The models also account for the influence of
2. State of the art
49
temperature, battery geometry and dimensions of their components (such as extension
of the cathode in order to reduce the risk of deposition of lithium on the cathode edges)
on battery performance.
Recent studies are focusing on the insertion of new species of ions such as sodium
and magnesium ions, and future research should focus on theoretical models for the
optimization of separators and electrodes for sodium and magnesium-ion batteries.
Regarding the implementation of the optimized models for electrodes, future
studies may focus on the use of different active materials and evaluate the influence of
electrical potential, porosity and capabilities of electrodes on insertion and extraction of
lithium, magnesium and sodium ions, in order to find more efficient electrodes. Future
work may focus on nano- and micro-porous electrode structures based on pure polymers
and nanocomposites, combining selected fillers with organic matrix.
Studies related to the separator membranes should improve knowledge on the
influence of the degree of porosity, pore size, the tortuosity, MacMullin number,
Bruggeman coefficient. The characteristics of the material for the separator, including
electrical insulation capacity (electrical properties), flexibility and mechanical stability
(mechanical properties), degree of degradation with the electrolyte, relative
performance against short circuits, ease of insertion into the electrolyte, effect of
thickness and ionic conductivity in battery performance should be further addressed.
Finally, once materials have been improved, charging characteristics, energy density
and discharge capacity of the batteries must be studied and models of Li, Mg and Na-
ion batteries should be optimized taking into account the influence of variables such as
temperature, pressure and geometry.
2. State of the art
50
2.5 References
1. Sikha, G., R.E. White, and B.N. Popov, A Mathematical Model for a Lithium-
Ion Battery/Electrochemical Capacitor Hybrid System. Journal of The
Electrochemical Society, 2005. 152(8): p. A1682-A1693.
2. Chirkov, Y.G., V.I. Rostokin, and A.M. Skundin, Computer simulation of
operation of lithium-ion battery: Galvanostatics, central problem of theory,
calculation of characteristics of thin active layers with low diffusion coefficients.
Russian Journal of Electrochemistry, 2011. 47(11): p. 1239-1249.
3. Subramanian, V.R., V. Boovaragavan, and V.D. Diwakar, Toward Real-Time
Simulation of Physics Based Lithium-Ion Battery Models. Electrochemical and
Solid-State Letters, 2007. 10(11): p. A255-A260.
4. Ramadesigan, V., et al., Efficient Reformulation of Solid-Phase Diffusion in
Physics-Based Lithium-Ion Battery Models. Journal of The Electrochemical
Society, 2010. 157(7): p. A854-A860.
5. Subramanian, V.R., et al., Mathematical Model Reformulation for Lithium-Ion
Battery Simulations: Galvanostatic Boundary Conditions. Journal of The
Electrochemical Society, 2009. 156(4): p. A260-A271.
6. Gomadam, P.M., et al., Mathematical modeling of lithium-ion and nickel battery
systems. Journal of Power Sources, 2002. 110(2): p. 267-284.
7. Garcı́a, R.E., et al., Microstructural Modeling and Design of Rechargeable
Lithium-Ion Batteries. Journal of The Electrochemical Society, 2005. 152(1): p.
A255-A263.
8. Sikha, G., B.N. Popov, and R.E. White, Effect of Porosity on the Capacity Fade
of a Lithium-Ion Battery: Theory. Journal of The Electrochemical Society, 2004.
151(7): p. A1104-A1114.
9. Ferguson, T.R. and M.Z. Bazant, Nonequilibrium Thermodynamics of Porous
Electrodes. Journal of The Electrochemical Society, 2012. 159(12): p. A1967-
A1985.
10. Renganathan, S., et al., Theoretical Analysis of Stresses in a Lithium Ion Cell.
Journal of The Electrochemical Society, 2010. 157(2): p. A155-A163.
2. State of the art
51
11. Dao, T.-S., C.P. Vyasarayani, and J. McPhee, Simplification and order reduction
of lithium-ion battery model based on porous-electrode theory. Journal of Power
Sources, 2012. 198(0): p. 329-337.
12. Lee, J.-W., Y.K. Anguchamy, and B.N. Popov, Simulation of charge–discharge
cycling of lithium-ion batteries under low-earth-orbit conditions. Journal of
Power Sources, 2006. 162(2): p. 1395-1400.
13. C++. http://www.cplusplus.com/. Available from: http://www.cplusplus.com/.
14. MatLab. http://www.mathworks.com/products/matlab/. Available from:
http://www.mathworks.com/products/matlab/.
15. Simulink. http://www.mathworks.com/products/simulink/. Available from:
http://www.mathworks.com/products/simulink/.
16. Fluent. http://www.ansys.com/. Available from: http://www.ansys.com/.
17. Studio, B.D. http://www.batterydesignstudio.com/. Available from:
http://www.batterydesignstudio.com/.
18. Multiphysics, C. http://www.comsol.com/. Available from:
http://www.comsol.com/.
19. Doyle, M., T.F. Fuller, and J. Newman, Modeling of Galvanostatic Charge and
Discharge of the Lithium/Polymer/Insertion Cell. Journal of The
Electrochemical Society, 1993. 140(6): p. 1526-1533.
20. Doyle, M., et al., Comparison of Modeling Predictions with Experimental Data
from Plastic Lithium Ion Cells. Journal of The Electrochemical Society, 1996.
143(6): p. 1890-1903.
21. Martinez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
22. Harb, J.N. and R.M. LaFollette, Mathematical Model of the Discharge Behavior
of a Spirally Wound Lead‐Acid Cell. Journal of The Electrochemical Society,
1999. 146(3): p. 809-818.
23. Long, J.W., et al., Three-Dimensional Battery Architectures. Chemical Reviews,
2004. 104(10): p. 4463-4492.
24. Evitts, G.F.K.a.R.W., Two-Dimensional Lithium-Ion Battery Modeling with
Electrolyte and Cathode Extensions,. Advances in Chemical Engineering and
Science, 2012. 2 No. 4: p. 423-434.
2. State of the art
52
25. E. Scott, G.T., B. Anderson and C. Schmidt, Anoma- lous Potentials in Lithium
Ion Cells: Making the Case for 3-D Modeling of 3-D Systems. The
Electrochemical So- ciety Meeting, Orlando, 13 October 2003.
26. E. Scott, G.T., B. Anderson and C. Schmidt, Observation and Mechanism of
Anomalous Local Potentials during Charging of Lithium Ion Cells. The
Electrochemical Society Meeting, Paris, 29 April 2003.
27. West, K., T. Jacobsen, and S. Atlung, Modeling of Porous Insertion Electrodes
with Liquid Electrolyte. Journal of The Electrochemical Society, 1982. 129(7):
p. 1480-1485.
28. Fuller, T.F., M. Doyle, and J. Newman, Simulation and Optimization of the Dual
Lithium Ion Insertion Cell. Journal of The Electrochemical Society, 1994.
141(1): p. 1-10.
29. Tang, M., P. Albertus, and J. Newman, Two-Dimensional Modeling of Lithium
Deposition during Cell Charging. Journal of The Electrochemical Society, 2009.
156(5): p. A390-A399.
30. Eberman, K., et al., Material and Design Options for Avoiding Lithium Plating
during Charging. ECS Transactions, 2010. 25(35): p. 47-58.
31. Arora, P., et al., Comparison between computer simulations and experimental
data for high-rate discharges of plastic lithium-ion batteries. Journal of Power
Sources, 2000. 88(2): p. 219-231.
32. Park, M., et al., A review of conduction phenomena in Li-ion batteries. Journal
of Power Sources, 2010. 195(24): p. 7904-7929.
33. Wolf, H., et al., Carbon–fiber–silicon-nanocomposites for lithium-ion battery
anodes by microwave plasma chemical vapor deposition. Journal of Power
Sources, 2009. 190(1): p. 157-161.
34. Yu, S., et al., Investigation of design parameter effects on high current
performance of lithium-ion cells with LiFePO4/graphite electrodes. Journal of
Applied Electrochemistry, 2012. 42(6): p. 443-453.
35. Chen, Y.H., et al., Porous cathode optimization for lithium cells: Ionic and
electronic conductivity, capacity, and selection of materials. Journal of Power
Sources, 2010. 195(9): p. 2851-2862.
2. State of the art
53
36. Sikha, G. and R.E. White, Analytical Expression for the Impedance Response
for a Lithium-Ion Cell. Journal of The Electrochemical Society, 2008. 155(12):
p. A893-A902.
37. Yusof, N., A.S.H. Basari, and N.K. Ibrahim. Prediction of the lithium-ion cell
performance via concentration profile simulation. in Green and Ubiquitous
Technology (GUT), 2012 International Conference on. 2012.
38. De, S., et al., Model-based simultaneous optimization of multiple design
parameters for lithium-ion batteries for maximization of energy density. Journal
of Power Sources, 2013. 227(0): p. 161-170.
39. Cooper, S.J., et al., Image based modelling of microstructural heterogeneity in
LiFePO4 electrodes for Li-ion batteries. Journal of Power Sources, (0).
40. Nelson, G.J., et al., Microstructural Effects on Electronic Charge Transfer in Li-
Ion Battery Cathodes. Journal of The Electrochemical Society, 2012. 159(5): p.
A598-A603.
41. Ebner, M., et al., Tortuosity Anisotropy in Lithium-Ion Battery Electrodes.
Advanced Energy Materials, 2014. 4(5): p. n/a-n/a.
42. Jung, S., Mathematical model of lithium-ion batteries with blended-electrode
system. Journal of Power Sources, 2014. 264(0): p. 184-194.
43. Costa, C.M., M.M. Silva, and S. Lanceros-Mendez, Battery separators based on
vinylidene fluoride (VDF) polymers and copolymers for lithium ion battery
applications. RSC Advances, 2013.
44. Patel, K.K., J.M. Paulsen, and J. Desilvestro, Numerical simulation of porous
networks in relation to battery electrodes and separators. Journal of Power
Sources, 2003. 122(2): p. 144-152.
45. Sahapatsombut, U., H. Cheng, and K. Scott, Modelling of electrolyte
degradation and cycling behaviour in a lithium–air battery. Journal of Power
Sources, 2013. 243(0): p. 409-418.
46. Cai, L. and R.E. White, Mathematical modeling of a lithium ion battery with
thermal effects in COMSOL Inc. Multiphysics (MP) software. Journal of Power
Sources, 2011. 196(14): p. 5985-5989.
47. Srinivasan, V. and C.Y. Wang, Analysis of Electrochemical and Thermal
Behavior of Li-Ion Cells. Journal of The Electrochemical Society, 2003. 150(1):
p. A98-A106.
2. State of the art
54
48. Sakti, A., et al., A validation study of lithium-ion cell constant c-rate discharge
simulation with Battery Design Studio®. International Journal of Energy
Research, 2012: p. n/a-n/a.
49. Peng, R. and M. Pedram, An analytical model for predicting the remaining
battery capacity of lithium-ion batteries. Very Large Scale Integration (VLSI)
Systems, IEEE Transactions on, 2006. 14(5): p. 441-451.
50. DualFoil. http://www.cchem.berkeley.edu/jsngrp/fortran.html. Available from:
http://www.cchem.berkeley.edu/jsngrp/fortran.html.
51. Fang, W., O.J. Kwon, and C.-Y. Wang, Electrochemical–thermal modeling of
automotive Li-ion batteries and experimental validation using a three-electrode
cell. International Journal of Energy Research, 2010. 34(2): p. 107-115.
52. Gerver, R.E. and J.P. Meyers, Three-Dimensional Modeling of Electrochemical
Performance and Heat Generation of Lithium-Ion Batteries in Tabbed Planar
Configurations. Journal of The Electrochemical Society, 2011. 158(7): p. A835-
A843.
53. Bae, S., et al., Quantitative performance analysis of graphite-LiFePO4 battery
working at low temperature. Chemical Engineering Science, 2014. 118: p. 74-
82.
3. Methodology and Theoretical Models
55
3. Simulation of Lithium-ion Batteries: Methodology and
Theoretical Models
This chapter describes the methodology implemented in the simulations performed
in the different studies presented in the various chapters. The theoretical models used in
these simulations, such as the electrochemical and thermal models, are presented.
3. Methodology and Theoretical Models
57
3.1 Simulation of lithium-ion batteries
3.1.1 Methodology
The simulation intends to reproduce the main phenomena and processes of the
system under study based on physical, chemical and mathematical models. So, it is
important to understand the equations governing the phenomena and processes that
occurs in the different components of the battery, including electrodes,
electrolyte/separator and current collectors, Thus, it is important to identify the
appropriate theoretical models to describe the operation of lithium-ion batteries.
The main four steps that should be followed to implement a consistent simulation of
the battery are:
First step: Perform the state of the art on the work about modeling and
simulation of lithium-ion batteries that is present in the literature;
Second Step: Study and understand the physical and electrochemical
equations that describe the lithium-ions battery operation;
Third Step: Implementation of the model by finite element method (FEM)
through commercial software or programming language such as C++,
Matlab etc. Input of l the partial and ordinary differential equations (PDA
and ODE) in the software;
Fourth Step: Run the simulation and analyze the obtained data.
Figure 3.1 summarizes these steps.
3. Methodology and Theoretical Models
58
Figure 3.1 - Steps for the implementation of the simulations.
3.1.2 Development and execution of the simulation
This section presents the different phases that should be followed in the
construction of a simulation after identifying the theoretical models that will be applied.
In first the phase of construction of a simulation it is necessary to define the dimension
(1D, 2D and 3D) of the model to be applied in the lithium-ion battery, as shown in
figure 3.2. The battery can be represented in 1D, 2D and 3D. If the battery is developed
in 1D, only the values of several physical variables along the x-axis will be measured.
Thus, the three components of the battery (electrodes, separator/electrolyte and current
collectors) are represented by a line, as shown in figure 3.2a). In relation to the 2D
representation of the battery, the physical variables in the xx and yy axes will be
measured, i.e., the values of physical quantities will be obtained in different points with
the coordinates (x, y) of the battery. The battery components are defined as planes, as
3. Methodology and Theoretical Models
59
shown in figure 3.2b). Finally, in the 3D representation, the physical quantities are
obtained in the (coordinates (x, y, z)) points of the battery. In this case, the battery
components are represented by volumes, as shown in Figure 3.2.c).
Figure 3.2 - Representation of the dimension of the battery for the application of the
theoretical model: a) 1D, b) 2D and c) 3D.
Then, it is necessary to draw the geometry of the battery and its components
(collectors, electrodes and separator/electrolyte), as shown in figure 3.3. Different
geometries can be defined for lithium ion batteries according to the objective of the
study.
3. Methodology and Theoretical Models
60
Figure 3.3 - Design of different geometries for lithium-ion batteries.
The next phase is characterized by the introduction of the equations governing the
phenomena that occur in the various components of the battery (collectors, electrodes,
separator/electrolyte). Then, it is important to define the active materials of electrodes
and electrolyte/separator. In this phase, all parameters and physical, chemical and
electrochemical constants of the materials of the battery components will be introduced.
Also, the boundary conditions and the initial values of the different variables should be
defined.
Once the physical and electrochemical quantities are measured at different battery
locations, it is necessary to define the mesh. The mesh should be defined according to
the dimensions of the battery.
Figure 3.4 shows that the mesh can be normal, fine or extremely fine. When the
mesh is extremely fine, it will be required a higher computational performance to
obtained the results. In contrast, if the mesh is normal, the element size of the mesh can
be large and the obtained results will show a larger associated error. Thus, the choice of
the element size of the mesh should take into account the order of magnitude of the
dimensions of the simulated battery. Typically, the element size of the mesh has an
order of magnitude below the order of magnitude of the battery.
3. Methodology and Theoretical Models
61
Figure 3.4 - Different size of the mesh: extremely fine, fine and normal.
After defining the size of the mesh of the battery, the study to be performed should
be selected: time dependent or stationary. Finally, the simulation will be performed. The
possible solutions from the electrochemical and thermal models of lithium-ion batteries
will be determined and the plots (1D, 2D or 3D) of the relevant parameters obtained
according to the objective of the study. Some examples of plots are: discharge curves
(delivery capacity), Nyquist plot, electrolyte salt concentration, solid lithium
concentration, electrolyte and electrode potential, electrode and electrolyte current
density, temperature, total heat generation rate of battery components, total ohmic heat
generation rate of battery components, total reversible heat generation rate of electrodes
and total reaction heat generation rate of electrodes as a function of time or space.
3. Methodology and Theoretical Models
62
3.2 Theoretical models of lithium-ion batteries: Electrochemical and Thermal
models
The simulations developed in this thesis are based on theoretical models that
govern the phenomena that occur in the different components of the lithium ion battery
(electrodes, separator/electrolyte and current collectors). The theoretical models applied
in the simulations are the electrochemical and thermal models.
As the electrochemical and thermal models are constituted by partial and ordinary
differential equations, the numerical resolution method used in the simulations was
based on the Finite Element Method. The batteries were simulated in 1 and 2
dimensions. The choice of these dimensions (1D and 2D) for the simulations is due to
the computational efficiency. The application of 3-dimensional models increases
substantially the number of points of the space to be measured and decreases
computational performance. Further, the 1D and 2D simulations properly represent the
performance of the battery.
The simulations developed on the various batteries are based on the electrochemical
model. The electrochemical model is based on the Doyle/Newman model [1-3], which
shows all the physical, chemical and electrochemical phenomena associated with the
operation of lithium ion batteries. When the thermal model is introduced, the aim is to
account the heats produced by the battery in its operation, taking into account the
corresponding thermal equations [4].
Then, the fundamental equations of the electrochemical and thermal models are
presented in table 3.1. These equations are applied to the different components of the
battery (electrodes, separator and current collectors), as shown in table 3.1.
Table 3.1 - Equations governing various phenomena within a battery [1-4].
Electrochemical model (Newman/Doyle/Fuller)
Physical process Governing Equation Region
Solid phase
conduction
Li
EiefFaj
x 2
2
2
,
Electrodes
Electrolyte phase
conduction
2
20
2
2
2
, ln1
2
x
Ct
F
kRTFaj
x
KL
LiLief
Electrodes,
Separator
3. Methodology and Theoretical Models
63
Electrolyte phase
diffusion 0
2
2
2
,1
ta
x
CD
t
C LiefLi
Electrodes,
Separator
Solid phase
diffusion
r
C
rr
CD
t
C EELi
E 22
2
Electrodes
Activation reaction
(Butler-Volmer
Kinetics)
iLERriERriEiEiLi URT
Fcccckj
ii
5.0
sinh2 5.05.0
,
5.0
,max,,
Electrodes
Diffusion/ionic and
electronic
conductuvity
5.1,, bruggkk brugg
ilief
Electrodes,
Separator
5.1,, bruggDD brugg
ilief
Electrodes,
Separator
cfcccef ,, 1
Electrodes
Specific interfacial
area
cfc
i
cR
a ,13
Electrodes
Thermal model
Physical process Governing Equation
Region
Energy balance itotaliiipi Qy
T
x
T
dt
dTC ,2
2
2
2
,
Electrodes,
Separator
Total heat
generation rate of
the electrodes
pniQQQQ iohmicireversibleireactionitotal ,,,,,,
Electrodes
Total heat
generation rate of
the separator
siQQ iohmicitotal ,,,
Separator
Total reaction heat
generation rate pniUFaJQ LEireaction ,,,
Electrodes
Total reversible
heat generation rate pni
T
UFaJTQ ireversible ,,,
Electrodes
Total ohmic heat
generation rate of
the electrodes
pniyy
ct
F
RTk
xx
ct
F
RTk
yk
xk
yxQ
iiefiief
iefiefiefiefiohmic
,,ln
12ln
12
2,2,
2
2,
2
2,
2
1,
2
1,,
Electrodes
3. Methodology and Theoretical Models
64
Total ohmic heat
generation rate of
the separator
siyy
ct
F
RTk
xx
ct
F
RTk
yk
xkQ
iief
iief
iefiefiohmic
,ln
12
ln1
2
2,
2,
2
2,
2
2,,
Separator
Temperature
dependent open
circuit potential of
the electrode
pnidT
dUTTUU refirefi ,,,
Electrodes
Heat flux transfer
between the battery
and the external
environment
psniTThT externali ,,,
All boundaries
between the
battery and the
external
environment
Auxiliary equations:
a) Ionic conductivity as a function of temperature [5]:
ki(T) = c (-10.5+(0.0740T)-((6.9610-5) (T2))+(0.668c)-
-(0.0178cT)+((2.810-5)c (T2))+(0.494c2)-((8.8610-4) (c2)*(T)))2
b) Reaction rate coefficient of the electrodes as a function of temperature [5]:
Kt,i (T)= kt298,15,i exp(-(Eak,i/R) (1/T-1/298,15))
c) Diffusion coefficient of the salt in the electrolyte as a function of temperature
[5]:
Di(T) = 10^(-(0.22c)-4.43-((54)/(T-229-(5c))))
d) Diffusion coefficient of Li ions in the electrode as a function of temperature [6]:
DLI(T) = DLI exp(-(Ead,i/R) (1/T-1/298,15))
The boundary conditions, parameters and initial values are defined according to the
objective of the study, so they will be presented in each study developed in chapters 4,
5, 6, 7 and 8.
3. Methodology and Theoretical Models
65
3.3 References
1. Doyle, M., T.F. Fuller, and J. Newman, Modeling of Galvanostatic Charge and
Discharge of the Lithium/Polymer/Insertion Cell. Journal of The
Electrochemical Society, 1993. 140(6): p. 1526-1533.
2. Doyle, M., et al., Comparison of Modeling Predictions with Experimental Data
from Plastic Lithium Ion Cells. Journal of The Electrochemical Society, 1996.
143(6): p. 1890-1903.
3. Fuller, T.F., M. Doyle, and J. Newman, Simulation and Optimization of the
Dual Lithium Ion Insertion Cell. Journal of The Electrochemical Society, 1994.
141(1): p. 1-10.
4. Cai, L. and R.E. White, Mathematical modeling of a lithium ion battery with
thermal effects in COMSOL Inc. Multiphysics (MP) software. Journal of Power
Sources, 2011. 196(14): p. 5985-5989.
5. Gerver, R.E. and J.P. Meyers, Three-Dimensional Modeling of Electrochemical
Performance and Heat Generation of Lithium-Ion Batteries in Tabbed Planar
Configurations. Journal of The Electrochemical Society, 2011. 158(7): p. A835-
A843.
6. Bae, S., et al., Quantitative performance analysis of graphite-LiFePO4 battery
working at low temperature. Chemical Engineering Science, 2014. 118: p. 74-
82.
4. Modelling of the separator membranes
67
4. Modelling separator membranes physical
characteristics for optimized lithium ion battery
performance
This chapter presents the evaluation of the influence of different geometrical
parameters of the separator in the performance of lithium ion batteries. The effect of
varying separator membrane physical parameters such as degree of porosity, tortuosity
and thickness, on battery delivered capacity was studied. This was achieved by a
theoretical mathematical model relating the Bruggeman coefficient with the degree of
porosity and tortuosity. The ionic conductivity of the separator and consequently the
delivered capacity values obtained at different discharge rates depends on the value of
the Bruggeman coefficient, which is related with the degree of porosity and tortuosity of
the membrane.
This chapter is based on the following publication:
“Modeling separator membranes physical characteristics for optimized lithium ion
battery performance”, D. Miranda, C.M. Costa, A.M. Almeida, S. Lanceros-Méndez,
Solid State Ionics 278 (2015) 78-84.
4. Modelling of the separator membranes
69
4.1 Introduction
Taking into account the rapid technological advances in portable electronic devices,
such as mobile-phone, computers, e-labels, e-packaging and disposable medical testers,
there is an increasing need for improving the autonomy and performance of batteries
independently of the battery type [1]. One of the types of the battery with the best
properties is the Lithium ion batteries, as they are lighter, cheaper, with higher energy
density (210Wh kg-1), no memory effect, prolonged service-life and higher number of
charge/discharge cycles when compared to other battery solutions [2].
In order to improve the autonomy and performance of lithium-ion batteries it is
necessary new advances in novel materials for improved delivery capacity, lifetime and
safety [3, 4].
In all battery devices, the separator membrane is located between the anode and
cathode and its main function is transferring the charge and prevent over-potential [5,
6].
The main characteristics of separator membranes for lithium ion batteries are
thickness, permeability, porosity and pore size, wettability by liquid electrolyte,
mechanical and dimensional stability [7, 8].
The separator is typically constituted by a polymer matrix, in which the membrane
is soaked by the electrolyte solution, i.e, salts are dissolved in solvents, water or organic
molecules.
For optimizing separator and electrodes materials (cathode and anode) it is essential
and critical the use of computer simulations of the battery performance [9].
These computer simulations are based on mathematical models that take into
account the physico-chemical properties of the materials to be used as electrodes and
separators, the organic solvents for electrolytes, and the geometry and dimensions of the
battery components [10, 11].
The computer simulation of the separator/electrolyte includes the correlation of
ionic conductivity of the polymeric membrane and the conductivity of the electrolyte
solution. Also the effective diffusivity is related to the Bruggeman coefficient. This
correlation is described through the Bruggeman equation which reflects the importance
of porosity and tortuosity of the material [12], the Bruggeman exponent being 1.5 for
ideal electrodes [7] and 2.4 at 4.5 for different electrolyte solution and polymer
membranes [13, 14]. In relation of electrodes materials, experimental results indicate
4. Modelling of the separator membranes
70
that the complexity of the porous electrodes induces tortuosity values that greatly
deviate from the classical Bruggeman ideal [15].
For same degree of porosity and polymeric membrane, was revealed through of the
utilization the different salts (LiBF4, LiTFSI and etc) in electrolyte solution that
tortuosity value varies between 3.3 at 4.1 [16].
In this work [17], the Bruggeman parameters for the commercial separators
membranes differ from the parameters reported in previous studies of separator
tortuosity.
It has been proven, on the other hand, that this exponent in not valid for real
electrodes or separator materials [12]. This is mainly due to effects in the separators that
are typically not accounted for. In this way, diffusion limitations in thick cells have
been reported [13], which become more prominent as the thickness of the electrodes
increases.
It is thus necessary for a proper description of separator performance, to take into
account the morphology parameters of separators that are important for the performance
of separator membranes such as porosity, pore size, tortuosity and thickness [18].
Figure 4.1 - Schematic representation of the main structure of a lithium ion battery.
The relevance of this work is to include these parameters in the computer
simulation models in order to optimize and improve battery performance.
A finite element method simulation has been thus carried out by in order to
quantitatively evaluate the effects of the dimensions of separator, porosity and tortuosity
towards optimization of its performance in lithium-ion batteries for the same electrodes
(anode and cathode) and independently of the electrolyte solution.
4. Modelling of the separator membranes
71
4.2 Theoretical model
4.2.1 General model
Anode, cathode and separator are the components of the lithium ion batteries
(Figure 4.1). Each constituent has a specific function in the operation of a lithium-ion
battery. The fundamental equations governing the main phenomena of the operation
process of a lithium-ion battery are based on the Doyle/Fuller/Newman model [19].
The Chapter 3 shows the main equations governing the different components of the
battery (cathode, anode and electrolyte/separator) and Table 4.1 shows the boundary
conditions applied in this study. The model takes into account all the variables
corresponding to the phenomena occurring in the electrodes and electrolyte/separator,
including: the diffusion and ionic conductivity of lithium ions in the electrolyte and
electrodes, the relation between the potential of the electrolyte and the local current
density on the electrodes (Ohm’s law), the relation between the potential of the
electrolyte and the local current density on electrolyte/separator (Ohm’s law), the
diffusion of lithium ions in the active material, the kinetics of the heterogeneous
reaction at the electrode/electrolyte interface, the open circuit voltage and the mass
transport flux (Chapter 3).
4. Modelling of the separator membranes
72
Table 4.1 - Boundary conditions applied in the simulation. The nomenclature is
indicated in the List of Symbols and Abbreviations.
Boundary Conditions
Electrodes (Anode and
cathode)
Cathode:
0
csa LLLx
L
x
C
sasa LLx
L
LLx
L
x
C
x
C
sasa LLx
Lcef
LLx
Lsef
x
CD
x
CD ,,
0,
psa LLLx
Lcef
x
CD
cef
TOTAL
LLLx
E I
xcsa ,
0,ELLxEsa
sasa LLx
Lcef
LLx
Lf
xk
xk
,
0,
csa LLLx
Lcef
xk
Anode:
aa Lx
L
Lx
L
x
L
x
C
x
C
x
C
,0
0
00
,
x
Lcef
x
CD
aa Lx
Lsef
Lx
Laef
x
CD
x
CD ,,
0 ,00
aLx
E
xEx
aa Lx
Lf
Lx
Laef
xk
xk
, ,
00
,
x
Laef
xk
4. Modelling of the separator membranes
73
Separator/Electrolyte
aa Lx
Lsef
Lx
Laef
x
CD
x
CD ,,
sasa LLx
Lcef
LLx
Lsef
x
CD
x
CD ,,
aa Lx
Lf
Lx
Laef
xk
xk
,
sasa LLx
Lcef
LLx
Lf
xk
xk
,
Active Material Li
Li
Rr
E
r
E
D
J
r
C
r
C
sp
,00
4.2.2 Separator
The effective conductivity of separator is described through of the following
equation:
p
slf . (1)
where f is the effective ionic conductivity of the polymer separator, l is the ionic
conductivity of the electrolyte, s is the porosity of separator and p is the Bruggeman
exponent.
Usually, the value of p is 1.5, as it is admitted that the separator pores show an ideal
shape [20].
For battery separators it has been shown that Bruggeman exponent ranges between
2.4 [13] and 4.5 [19].
One important parameter influencing the battery separator performance is the
tortuosity (), which is defined by the ratio between the effective capillarity length
and the thickness of the sample [21]:
2
s
lf (2)
4. Modelling of the separator membranes
74
Taking account the equation 2, the tortuosity value is related with the ionic
transport and provides information about pore blockage which describes the average
pore connectivity of a solid. The ideal value of tortuosity is 1 for an ideal porous body
with cylindrical and parallel pores.
By relating equations 1 and 2, one obtains
s
p
ln
ln1
2
(3)
which shows how the Bruggeman exponent depends on the values of the tortuosity and
the porosity.
Also the salt diffusion coefficient is described through the following equation:
pslf DD (4)
where p is determined by equation 3.
4.3 Parameters and simulation model
The finite element method simulation implemented in this work is based on the
mathematical model of Newman group [22], considering the electrochemical and
transport processes in a 1D lithium ion battery structure consisting on a [positive
electrode | separator | negative electrode]. The equations describing the electrochemical
and transport processes of the separator were modified to include equation 3. The values
of the ionic conductivity and porosity included in this simulation model are the ones of
the P(VDF-TrFE) copolymer [16, 23, 24]. The choice of this separator/electrolyte is due
to its high ionic conductivity at room temperature, and very stable in function of
temperature, good mechanical properties and excellent electrochemical stability up to
4V [16, 23]. The values of the parameters used for each component of the battery are
listed in Table 4.2.
4. Modelling of the separator membranes
75
Table 4.2 - Parameters used in the simulations.
Parameter Unit Anode (LixC6) Separator Cathode (LixMn2O4)
CE,i,0 mol/m3 14870 3900
CE,i,max mol/m3 26390 22860
CL mol/m3 1000
r m 12,510-6 810-6
Li m 10010-6 Ls 18310-6
i S/m (6,510-1) 0,3571,5 (6,510-1) s
p (6,510-1) 0,4441,5
Di m2/s (4,010-10) 0,3571,5 (4,010-10) sp (4,010-10) 0,4441,5
DLI m2/s 3,910-14 110-13
Brugg or p 1,5 p 1,5
f,i 0,172 0,259
i 0,357 s 0,444
i S/m 100 3,8
i1C
A/m2 17,5
F C/mol 96487
T K 298,15
R J/mol K 8,314
For the electrodes, the values of the different parameters are constant and are
presented in Table 4.2. Relatively to the parameters of the separator, the ones indicated
in the table are considered fixed, and thickness (Ls), Bruggeman exponent (p) and
porosity (s) were varied in the simulations. The discharge protocol is the continuous
current system where the voltage cut-off occurs around of 2.65V at room temperature.
For each effect studied, were realized 3 simulations with < 0.1% error due that errors
associated with the finite element solution of equations is minimized with care in the
physical configuration of the problem.
4. Modelling of the separator membranes
76
4.4 Results and Discussion
4.4.1 Effect of separator/electrolyte
The behavior of the battery at different scan rates for a battery including a polymer
porous membrane or free electrolyte is shown in figure 4.2.
0 2 4 6 8 10 12 14 16 18 20
2.5
3.0
3.5
4.0
a)
Voltage
/ V
Capacity / Ahm-2
0.15C
0.3C
0.7C
1C
2C
3C
5C
0 2 4 6 8 10 12 14 16 18 20
2.5
3.0
3.5
4.0
b)
Vo
lta
ge
/ V
Capacity / Ahm-2
0.15C
0.3C
0.7C
1C
2C
3C
5C
Figure 4.2 - Voltage as a function of delivered capacity at different scan rates for: a)
free electrolyte and b) battery separator membrane with 70% of porosity and 3.8 of
tortuosity.
Figure 4.2 shows the voltage as a function of delivered capacity for the free
electrolyte without membrane (figure 4.2a)) and for a porous membrane with 70% of
porosity and 3.8 of tortuosity (figure 4.2b)). Independently of electrolyte type (figure
4.2), as expected is observed, a progressive decrease of the discharge value with
increasing the current density due to the ohmic drop. This fact is observed in
experimental results but this simulation model assumes a constant value for the solid-
phase diffusion coefficient [18].
Figure 4.2 shows that for low (0.15C) and medium (2C) discharge rates there is no
variation in the results obtained for the delivered capacity of batteries with free
electrolyte or polymer separator membrane. The separator membrane, therefore, does
not influence negatively the performance of the battery up to medium discharge rates
(2C).
4. Modelling of the separator membranes
77
On the other hand, at high discharge rates (5C), the values of the delivered capacity
for the separator membranes are slightly smaller when compared with the samples with
free electrolyte. Thus, Figure 4.3 shows the delivered capacity as a function of the scan
rate for the aforementioned systems.
0 1 2 3 4 5
12
13
14
15
16
17
18
19
C
apa
city / A
hm
-2
Discharge current of each cycle / C
1C=17.5 A.m-2
Electrolyte
Separator membrane
Figure 4.3 - Delivered capacity as a function of the scan rate for free electrolyte and
separator membrane batteries.
Figure 4.3 shows that there are differences in the delivered capacity for both
systems for scan rates above 3C, the delivered capacity being slightly smaller for
battery systems with separator membranes.
This is due to the fact that, for high discharge rates, the diffusion and mobility of
lithium ions should be larger in order to cross through the separator membrane. The
ionic conductivity of the separator membrane is lower in comparison to the free
electrolyte, which is reflected in the lower performance of the battery system with
separator membrane.
Although it is observed a decrease in the performance of the battery system with
separator membrane for the higher discharge rates, the differences in the delivered
capacity values between the separator membrane and the free electrolyte are not
significant. In this way, the introduction of a polymer membrane in the battery separator
will not strongly hinder the battery performance for low, medium and high battery
discharge rates.
4. Modelling of the separator membranes
78
4.4.2 Effect of the variation of separator membrane physical parameters on
battery performance
Considering that the inclusion of the polymer membrane in the separator does not
significantly affects the performance of the battery, the effect of the variation of relevant
physical parameter of the separator membrane such as degree of porosity, tortuosity and
thickness on battery performance will be addressed.
4.4.2.1 Degree of porosity
Figure 4.4 illustrates the effect of the degree of porosity on the voltage vs. capacity
characteristics for separator membranes with a fixed tortuosity value of 3.8 at low (0.15
C, figure 4.4a)) and high scan rate (5C, figure 4.4b)).
0 2 4 6 8 10 12 14 16 18 20
2.4
2.8
3.2
3.6
4.0
a)
Voltage
/ V
Capacity / Ahm-2
= 0.15
= 0.3
= 0.5
= 0.7
= 0.9
0 2 4 6 8 10 12
2.4
2.7
3.0
3.3
3.6
3.9
b)
Voltage
/ V
Capacity / Ahm-2
= 0.15
= 0.3
= 0.5
= 0.7
= 0.9
Figure 4.4 - Voltage as a function of delivered capacity for batteries with separator
membranes with different degrees of porosity with tortuosity of 3.8 at scan rates of a)
0.15C and b) 5C.
Figure 4.4a) shows that for 0.15C and degrees of porosity between 15% and 90%
there is no relevant variation in the performance of the battery system, the degree of
porosity not affecting therefore the performance of the battery.
4. Modelling of the separator membranes
79
On the other hand, for high discharge rates (figure 4.4b)) it is observed a strong
decrease in the battery performance for degrees of porosity below 50%, which further
decreases with decreasing degree of porosity.
For degrees of porosity above 50%, the delivered capacity just slightly increases
with increasing the degree of porosity with capacity values between 11 Ah/m2 to 12
Ah/m2 for degrees of porosity between 50% and 90%.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
4
6
8
10
12
14
16
18
20
Ca
pa
city / A
hm
-2
Porosity
0.15C
2C
5C
Figure 4.5 - Delivered capacity as a function of the degree of porosity at different scan
rates: 0.15C, 2C and 5C.
Figure 4.5 shows the effect of the degree of porosity in the delivered capacity at
different scan rates, 0.15C, 2C and 5C, for separator membranes with tortuosity value of
3.8.
For the scan rate of 0.15C, the performance of the battery measured through the
delivered capacity is independent of the degree of porosity of the separator membrane.
For a scan rate of 2C, the delivered capacity increases with increasing degree of
porosity up to 30%, remaining constant for higher degrees of porosity.
However, at high scan rates, 5C, the delivered capacity increases strongly with
increasing degree of porosity up to 50%. Taking account this behavior and the results
shown in Figures 4.4 and 4.5, it is considered that good battery performances are
obtained for degrees of porosity above 50%. It is to notice that the degree of porosity is
correlated with the uptake value but also the affinity between salt and polymer chain
which in turn affects the ionic conductivity value of the separator [25]. On the other
4. Modelling of the separator membranes
80
hand, the ionic conductivity and transport occurs mainly in the amorphous region which
undergo swelling to accommodate the electrolyte but with mechanical integrity [26, 27].
The mechanical integrity depend the degree of porosity and pore size of the separator
membrane but also the degree of crystallinity [27].
4.4.2.2 Tortuosity
In the simulations above, the value of the tortuosity has been considered fixed for
all membranes. It is nevertheless important to have in mind that this is one of the most
important characteristics of a separator membrane, as the tortuosity value is correlated
with the ionic conductivity of the separator.
Figure 4.6 shows the effect of the different tortuosity values in the delivered
capacity of the battery system at different scan rates for different degrees of porosity.
0 4 8 12 16 20 24 28 32 3613
14
15
16
17
18
19
a)
Ca
pa
city / A
hm
-2
Tortuosity
=0.15
=0.3
=0.5
=0.7
=0.9
0 1 2 3 4 5 6 7 8 9 108
9
10
11
12
13
14
15
16
17
b)
Ca
pa
city / A
hm
-2
Tortuosity
=0.15
=0.3
=0.5
=0.7
=0.9
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0
2
4
6
8
10
12
14
c)
Ca
pa
city / A
hm
-2
Tortuosity
=0.15
=0.3
=0.5
=0.7
=0.9
Figure 4.6 - Delivered capacity as a function of tortuosity for membranes with different
degrees of porosity: a) low scan rate, 0.15C, b) moderate scan rate, 2C and c) high scan
rate, 5C.
4. Modelling of the separator membranes
81
Figure 4.6a) shows the limit values of tortuosity for each degree of porosity at
0.15C at which there is no decreases of the capacity value of the battery.
It is verified that for a degree of porosity of 15%, the limit tortuosity value is
around 14, but for a degree of porosity of 90% this value increases up to 33, followed
by a drastic decrease in the delivered capacity value.
A similar behavior is observed for the scan rates of 2C (figure 4.6b)) and 5C (figure
4.6c)), the main differences being the limit value of the tortuosity at the best
performance of the battery.
Table 4.3 shows the limit value of tortuosity for the different degrees of porosity
and scan rates, as obtained from figure 4.6.
Table 4.3 - Limit value of tortuosity for different degrees of porosity and scan rates.
0.15C 2C 5C
ε=0.15 12 2 1
ε=0.30 16 3.8 2
ε=0.50 24 4 2
ε=0.70 26 4 4
ε=0.90 30 6 5
Table 4.3 shows that for higher values of porosity, the limit value of tortuosity, at
which a constant delivered capacity is maintained, increases.
Thus for a given degree of porosity and a discharge rate it is observed that the
tortuosity has a limit value to maintain a good performance of the separator and,
consequently, a good battery performance.
After the limit value of tortuosity for a given degree of porosity it is observed a
significant decrease in the delivered capacity, strongly decreasing the performance of
the battery.
It is observed that for higher values of the degree of porosity, the limit value of
tortuosity can be higher without affecting the performance of battery as it is reflected in
equation 2. As the capacity of the battery is related with the ionic conductivity of the
separator/electrolyte, increasing the value of the tortuosity for a given degree of porosity
and discharge rate results in a decrease of the ionic conductivity decreases, leading to a
reduction in capacity [25, 26].
4. Modelling of the separator membranes
82
4.4.2.3 Dimension/thickness
Another important parameter of the separator is its thickness. Figure 4.7 shows the
voltage vs delivered capacity, at 0.15C and 5C scan rates, of a battery as a function of
the separator membrane thickness for a membrane with 70% of porosity and 3.8 of
tortuosity knowing that separator membrane presents mechanical integrity. The
mechanical integrity depends the degree of porosity but also pore size.
0 5 10 15 20
2.4
2.8
3.2
3.6
4.0
4.4
a)
Vo
lta
ge
/ V
Capacity / Ahm-2
Ls=210m
Ls=200m
Ls=150m
Ls=100m
Ls=52m
Ls=32m
Ls=1m
0 1 2 3 4 5 6 7 8 9 10 11 12 13
2.4
2.8
3.2
3.6
4.0b)
Vo
lta
ge
/ V
Capacity / Ahm-2
Ls=210m
Ls=200m
Ls=150m
Ls=100m
Ls=52m
Ls=32m
Ls=1m
Figure 4.7 - Voltage as a function of the delivered capacity for battery separator
membranes with different thicknesses, 70% of porosity and 3.8 of tortuosity: a) 0.15C
and b) 5C.
Figure 4.7a) shows that the thickness of the separator membrane does not affect the
delivered capacity value of the battery system for a scan rate of 0.15C. This behavior is
not observed for higher scan rates (figure 4.7b)), 5C) in which increasing the thickness
of the separator leads to a decrease of the delivered capacity.
This fact is correlated with the longer path that lithium ions have to move through
the separator membrane which leads to a decreasing delivered capacity value.
Figure 4.8 presents the delivered capacity as a function of the thickness of separator
at different scan rates.
4. Modelling of the separator membranes
83
0 20 40 60 80 100 120 140 160 180 200 2204
6
8
10
12
14
16
18
20
Ca
pa
city / A
hm
-2
Thickness of separator / m
0.15C
2C
5C
Figure 4.8 - Delivered capacity as a function of the separator thickness at different scan
rates: 0.15C, 2C and 5C.
Figure 4.8 shows that the ideal value of the separator thickness is between 1μm and
32μm, leading to proper battery capacity values independently of the scan rate.
Based on these results (figure 4.8), it is concluded that according to the polymer
membrane used in the separator with a degree of porosity and tortuosity value assigned,
it is possible to determine the maximum thickness value possible for which there is not
a decrease of battery capacity for each discharge rate applied.
Normally, it is observed commercial values of thickness of separator membrane and
degree of porosity that are < 25μm and 40-70% in which these values are attributed for
separator but without referring the importance and the influence of tortuosity value in
the separator membrane once that these parameters influence its ionic conductivity
value [7, 20]. It is observed that the ideal thickness depends also on the discharge scan
rate. Taking into account these results, the thickness value depends on the degree of
porosity and tortuosity of the separator membrane.
With alteration of these parameters (degree of porosity and tortuosity value), it will
be obtained new maximum thickness values of separator without decreasing the
delivered capacity value of battery.
4. Modelling of the separator membranes
84
4.5 Conclusions
Separator membranes are essential to obtain good performance of lithium-ion
batteries. In this way it is required the optimization of separator parameters such as
porosity, tortuosity and thickness taking into account the delivered capacity value of the
battery.
Based on a mathematical model that describes the electrochemical and ionic
transport processes within the separator, variables such as degree of porosity and
tortuosity were included through the Bruggeman exponent at different scan rates.
The Bruggeman coefficient, which depends on the degree of porosity and
tortuosity, has a strong influence on the values of the diffusion coefficient and ionic
conductivity of lithium ions in the separator and, consequently, on the delivered
capacity of the battery. The inclusion of the separator membrane in the simulation
model of the battery system does not affect the performance of the battery in
comparison to the free electrolyte without polymer membrane. It was then demonstrated
the existence of optimal values of the degree of porosity and tortuosity. Independently
of the scan rate, the ideal value of the degree of porosity is above 50% and the separator
thickness should range between 1μm and 32μm with mechanical integrity in order to
maintain good battery performance.
4. Modelling of the separator membranes
85
4.6 References
1. Wakihara, M. and O. Yamamoto, Lithium ion batteries: fundamentals and
performance1998: Kodansha.
2. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
3. Marom, R., et al., A review of advanced and practical lithium battery materials.
Journal of Materials Chemistry, 2011. 21(27): p. 9938-9954.
4. Etacheri, V., et al., Challenges in the development of advanced Li-ion batteries:
a review. Energy & Environmental Science, 2011. 4(9): p. 3243-3262.
5. Yoshio, M., R.J. Brodd, and A. Kozawa, Lithium-Ion Batteries: Science and
Technologies2010: Springer.
6. Manuel Stephan, A., Review on gel polymer electrolytes for lithium batteries.
European Polymer Journal, 2006. 42(1): p. 21-42.
7. Arora, P. and Z. Zhang, Battery Separators. Chemical Reviews, 2004. 104(10):
p. 4419-4462.
8. Huang, X., Separator technologies for lithium-ion batteries. Journal of Solid
State Electrochemistry, 2011. 15(4): p. 649-662.
9. Ramadesigan, V., et al., Modeling and Simulation of Lithium-Ion Batteries from
a Systems Engineering Perspective. Journal of The Electrochemical Society,
2012. 159(3): p. R31-R45.
10. Martínez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
11. Miranda, D., C.M. Costa, and S. Lanceros-Mendez, Lithium ion rechargeable
batteries: State of the art and future needs of microscopic theoretical models and
simulations. Journal of Electroanalytical Chemistry, 2015. 739: p. 97-110.
12. Patel, K.K., J.M. Paulsen, and J. Desilvestro, Numerical simulation of porous
networks in relation to battery electrodes and separators. Journal of Power
Sources, 2003. 122(2): p. 144-152.
13. Arora, P., et al., Comparison between computer simulations and experimental
data for high-rate discharges of plastic lithium-ion batteries. Journal of Power
Sources, 2000. 88(2): p. 219-231.
4. Modelling of the separator membranes
86
14. Doyle, M., et al., Comparison of Modeling Predictions with Experimental Data
from Plastic Lithium Ion Cells. Journal of The Electrochemical Society, 1996.
143(6): p. 1890-1903.
15. Ding-Wen, C., et al., Validity of the Bruggeman relation for porous electrodes.
Modelling and Simulation in Materials Science and Engineering, 2013. 21(7): p.
074009.
16. Costa, C.M., et al., Influence of different salts in poly(vinylidene fluoride-co-
trifluoroethylene) electrolyte separator membranes for battery applications.
Journal of Electroanalytical Chemistry, 2014. 727(0): p. 125-134.
17. Cannarella, J. and C.B. Arnold, Ion transport restriction in mechanically strained
separator membranes. Journal of Power Sources, 2013. 226: p. 149-155.
18. Costa, C.M., et al., Poly(vinylidene fluoride)-based, co-polymer separator
electrolyte membranes for lithium-ion battery systems. Journal of Power
Sources, 2014. 245(0): p. 779-786.
19. Yu, S., et al., Investigation of design parameter effects on high current
performance of lithium-ion cells with LiFePO4/graphite electrodes. Journal of
Applied Electrochemistry, 2012. 42(6): p. 443-453.
20. Tye, F.L., Tortuosity. Journal of Power Sources, 1983. 9(2): p. 89-100.
21. Doyle, M., T.F. Fuller, and J. Newman, Modeling of Galvanostatic Charge and
Discharge of the Lithium/Polymer/Insertion Cell. Journal of The
Electrochemical Society, 1993. 140(6): p. 1526-1533.
22. Costa, C.M., M.M. Silva, and S. Lanceros-Mendez, Battery separators based on
vinylidene fluoride (VDF) polymers and copolymers for lithium ion battery
applications. RSC Advances, 2013. 3(29): p. 11404-11417.
23. Martins, P., A.C. Lopes, and S. Lanceros-Mendez, Electroactive phases of
poly(vinylidene fluoride): Determination, processing and applications. Progress
in Polymer Science, 2014. 39(4): p. 683-706.
24. Michot, T., A. Nishimoto, and M. Watanabe, Electrochemical properties of
polymer gel electrolytes based on poly(vinylidene fluoride) copolymer and
homopolymer. Electrochimica Acta, 2000. 45(8–9): p. 1347-1360.
25. Idris, N.H., et al., Microporous gel polymer electrolytes for lithium rechargeable
battery application. Journal of Power Sources, 2012. 201: p. 294-300.
4. Modelling of the separator membranes
87
26. Rajendran, S., O. Mahendran, and T. Mahalingam, Thermal and ionic
conductivity studies of plasticized PMMA/PVdF blend polymer electrolytes.
European Polymer Journal, 2002. 38(1): p. 49-55.
27. Ferreira, J.C.C., et al., Variation of the physicochemical and morphological
characteristics of solvent casted poly(vinylidene fluoride) along its binary phase
diagram with dimethylformamide. Journal of Non-Crystalline Solids, 2015. 412:
p. 16-23.
5. Theoretical simulation of the cathode
89
5. Theoretical simulation of the optimal relationship
between active material, binder and conductive additive
for lithium-ion battery cathodes
This chapter describes the theoretical simulations that have been carried out to
evaluate the influence of active material, binder and conductive additive relative
contents on electrode performance at various discharge rates. The simulations were
performed by the finite element method applying the Doyle/Fuller/Newman model for
two different active materials, C-LiFePO4 and LiMn2O4, and the obtained results were
compared with experimental data.
This chapter is based on the following publication:
“Theoretical simulation of the optimal relationship between active material, binder
and conductive additive for lithium-ion battery cathodes”, D. Miranda, A. Gören, C. M.
Costa, M. M. Silva, A. M. Almeida, S. Lanceros-Méndez, submitted.
5. Theoretical simulation of the cathode
91
5.1 Introduction
The rapid technological development of mobile electrical applications lead to the
increasingly important question of how to store electrical energy in a more efficient way
[1]. Thus, energy storage is critical in modern society, the most used energy storage
system being batteries [2], particularly, rechargeable lithium-ion batteries, introduced to
the market in 1992 by Sony [3]. Lithium-ion batteries are of increasing importance as
power sources as they are lighter, cheaper, show higher energy density, lower self-
discharge, no memory effect, prolonged service-life, higher number of charge/discharge
cycles, environmental friendliness and higher safety when compared to other battery
technologies [4]. There are two main types of batteries, defined as primary and
secondary batteries, the latter being rechargeable [5, 6].
The main constituents of lithium-ion batteries are the cathode, anode and the
separator membrane [7] and the key issues are to improve specific energy, power, safety
and reliability [8]. For the various components of the batteries it is necessary to improve
the materials that constitute them, being particularly relevant the cathode, due to its
influence on the capacity of the battery [9].
Cathodes are typically constituted by a polymer binder, a conductive additive and an
active material, the most used active materials being lithium iron phosphate (LiFePO4),
lithium nickel manganese cobalt oxide (LiNiMnCoO2), lithium cobalt oxide (LiCoO2),
lithium nickel oxide (LiNiO2), lithium nickel cobalt aluminum (LiNiCoAlO2), lithium
titanate oxide (LiTiO2) and lithium manganese spinel oxide (LiMn2O4), among others
[9].
The key characteristics of the active materials include being easily reducible,
reacting with lithium in a reversible manner, being a good electronic conductor and
stable, i.e. not undergoing structural variations of degradation with the loading and
unloading of the battery [4].
Relevant parameters of the cathodes that affect their performance include active
mass loading, porosity, thickness and the relation between active material, binder and
conductive additive [10-13].
The electrode density depends on the maximum amount of active material,
including the lowest possible amount of binder and conductive additive to obtain proper
mechanical and electrical properties, respectively [14].
5. Theoretical simulation of the cathode
92
The width of the cathode determines two main factors: the quantity of active
material and the resistance to mass transport; finally, the porosity of the electrode
affects the effective conductivity and the resistance to mass transfer [15].
For C-LiFePO4 active material, more than 40 electrode formulations have been
reported for active material, binder and conductive additive, the highest amount of
active material reaching 95% and the lowest amount for binder and conductive additive
being 2% and 3%, respectively [14]. For C-LiFePO4 as active material, the minimum
and maximum relative percentage of each component in the electrode slurry has been
reported as 60 to 95% for the active material, from 2 to 25% for the binder and from 3
to 30% for the conductive additive [14]. Percolation is achieved for a volume fraction of
active material of 30% [16].
Taking into account the state of art, it is thus necessary the optimization of the
electrode composition, allowing the fabrication of high-quality lithium-ion battery
cathodes for applications such as printed batteries [17]. This optimization can be guided
by computer simulation of the performance of a battery, based on the electrochemical
reactions describing the physical-chemical properties of the materials to be used as
electrodes and separators [18].
Thus, this chapter is devoted to the optimization of the cathode formulation
relationship (active material, conductive material and binder) for two active materials
(C-LiFePO4 and LiMn2O4) taking also into account the porosity and electronic
conductivity. In this way, the study has focus in understanding the optimal relationship
of the cathode components for obtaining higher capacity, maintaining constant the
width of the battery. The theoretical simulation model was first validated with
experimental results.
5.2 Preparation and characterization of the cathodes
For the validation of the theoretical model, cathodes were first prepared and
characterized. C-LiFePO4 (LFP, Particle size: D10=0.2 μm, D50=0.5 μm and D90=1.9
μm), carbon black (Super P-C45), poly(vinylidene fluoride) (PVDF, Solef 5130) and N-
methyl-2-pyrrolidone (NMP) were acquired from Phostech Lithium, Timcal Graphite &
Carbon, Solvay and Fluka, respectively. LiMn2O4 (LMO) was synthesized via sol gel
method as indicated in [19]. The cathode was prepared by mixing LFP or LMO as
5. Theoretical simulation of the cathode
93
active materials, Super P, and the polymer binder in NMP solvent with a weight ratio of
80:10:10 (wt.%).
After complete dissolution of the polymer binder, small amounts of dried mixed
solid material (LFP/LMO and Super P) were added to the solution under constant
stirring at room temperature. The electrode slurry was maintained under stirring for 3
hours at 1000 rpm to obtain a good dispersion.
The electrode slurry was spread onto an aluminum foil and dried in air atmosphere
at 80 ºC in a conventional oven (ED 23 Binder). After complete evaporation of the
solvent, the cathodes were dried at 90 ºC in vacuum before being transferred into a
glove-box.
Two Swagelok type cells were assembled in the home-made argon-filled glove box:
metallic lithium (8 mm diameter) was used as anode material; Whatman glass
microfiber filters (grade GF/A) (10 mm diameter) was used as separator; 1M LiPF6 in
ethylene carbonate-diethyl carbonate (EC-DEC, 1:1 vol) or in ethylene carbonate-
dimethyl carbonate (EC-DMC, 1:1 vol) (Solvionic) were used as electrolyte and the
prepared LFP/LMO electrodes were used as cathodes (8 mm diameter).
Charge-discharge measurements were carried out at room temperature at different
current densities (C/10 and C/2) in the voltage range from 2.5 to 4.2 V for LFP and
from 3.5 to 4.2 V for LMO using a Landt CT2001A Instrument.
5.3 Theoretical simulation model and model parameters
Two lithium half-cell batteries were simulated with the different active materials for
the cathode as well as with the different electrolyte solutions. The lithium-ion half-cell
battery structure was [anode, (Li metallic) | separator, P(VDF-TrFE) soaked in 1M
LiTFSI in PC | cathode, (LFP) or (LMO)].
The main equations governing the operation of the different components of the half-
cell batteries (Chapter 3) are based on the Doyle/Fuller/Newman model [20-25] and the
finite element method was implemented for the theoretical simulations.
In this work, the influence of the relative percentages of the three components of the
cathode (binder, active material and carbon black) in the performance of the battery will
be evaluated. The variables introduced in the model will be the percentage of active
material, C1, binder, C2, and carbon black, C3, respectively. The relative percentage of
5. Theoretical simulation of the cathode
94
each component affects the value of the porosity (c) of the cathode as well as its
electronic conductivity (c). Thus, the porosity of the cathode is represented by [13, 26]:
L
D
C
D
C
D
CWL
c
3
3
2
2
1
1
(1)
where L is the thickness of the electrode, W is weight of the electrode per unit area and
D1, D2 and D3 are the densities of the active material, the binder and the conductive
additive, respectively.
Further, the electronic conductivity of the cathode is represented by [13, 26]:
3
23loglog
bPurec (2)
where is the measured electronic conductivity of the neat conductive additive.
The parameters 2 and 3 are described by the following equations:
TotalVD
m
2
22 (3)
TotalmCm 22 (4)
and
TotalVD
m
3
33 (5)
TotalmCm 33 (6)
In equation 2, log 3Pure and b are constants, so c depends on the ratio 2/3. Thus,
2
3
2
3
3
2
23
32
3
2
D
Dn
D
D
C
C
DC
DC
(7)
with
3
2
C
Cn (8)
The percentages of the active material, the binder and the carbon black, will be called
hereafter C1, C2 and C3, respectively.
Finally, the parameters used for the simulations of the half-cells are indicated in
Table 5.1.
The nomenclature of the aforementioned equations and tables is shown in the List
of Symbols and Abbreviations.
Pure3
5. Theoretical simulation of the cathode
95
Table 5.1 - Parameters used for the simulations of the Li/LFP and Li/LMO half-cells.
Li/LFP and Li/LMO cell Parameter Unit Electrolyte Cathode (LFP/LMO)
CE,i,0 mol/m3 800/3900
CE,i,max mol/m3 22806
CL mol/m3 1000
r m 1,710-6/1,510-6
Lc m 7010-6/9910-6
Ls m 43010-6
kef,i S/m (valuea))0,301,5/(valueb))0,351,5 (valuea)b))c 1,5
Di m2/s (3,010-10)0,301,5/(7,510-11)0,351,5 (3,010-10/7,510-11)c 1,5
DLI m2/s 810-18/110-16
ki mol/s.m2 310-13/210-11
Brugg or p 1,5 1,5
i 0,30/0,35 c
i S/m c
3Pure S/m 100 [27]
i1C
A/m2 8.66/7.96
C1 C1
C2 C2
C3 C3
D1 g/m3 3,34106/2,93106
D2 g/m3 1.765106
D3 g/m3 1.9106
W
g/m2 64,6/21.8
b 1
VTotalc m3 4,4510-9/7,2510-9
mTotalc g 4,1110-3/1,610-3
General parameters
Cut-off
voltage V 2,5/3.4
F C/mol 96487
T K 298,15
R J/mol K 8,314 0
t 0,363
Abat m2 6.3610-5/5,0210-5
Electrolyte LiPF6 in EC:DEC/ LiPF6 in EC:DMC
Inert filler PVDF a) Model fits: Ionic condutivity as a function of salt concentration for LiPF6 in EC:DEC
[22]:
b) Model fits: Ionic condutivity as a function of salt concentration for LiPF6 in EC:DMC
[28]: 44332224 10977,4101708,51082683,11032702,210905,2 cccckl
kl = 0,0911+1,910c-1,052c2 +0,1554c3
5. Theoretical simulation of the cathode
96
5.4 Results and discussion
In order to evaluate the optimal relationship between the active material, the
conductive material and the polymer binder within the cathodes in order to obtain
higher capacity values and optimal performance in lithium ion half-cells, theoretical
simulations were performed in two lithium ion half-cells (Li/LFP and Li/LMO) taking
into account the equations (Chapter 3 and 5.3) describing the phenomena associated to
battery performance. Thus, the delivery capacity and impedance for both half-cells was
obtained. Further, the electrode and electrolyte current density was also obtained for the
Li/LFP half-cells.
5.4.1 LFP and LMO half-cells: validation of the theoretical model
First, the simulation model was validated by comparing the theoretical and
experimental results obtained for the Li/LFP and Li/LMO half-cells (figures 5.1a) and
5.1b)).
Figures 5.1a) and 5.1b) show a comparison of the experimental results and the
simulation curves (full line) at C/10 and C/2 discharge rates for Li/LFP and Li/LMO,
respectively.
0 20 40 60 80 100 120 140 1602.4
2.6
2.8
3.0
3.2
3.4a)
130 140 150 160
2.6
2.8
3.0
3.2
3.4
Volta
ge / V
vs
Li/L
i+
Capacity / mAh.g-1
C10
C2
Voltage
/ V
vs L
i/Li+
Capacity / mAh.g-1
C10
C2
5. Theoretical simulation of the cathode
97
Figure 5.1 - Voltage as a function of the delivered capacity at C/10 and C/2 discharge
rates for the a) Li/LFP and b) Li/LMO half-cells.
For both half-cells and discharge rates, a good agreement is observed between the
theoretical and experimental values, validating therefore the simulation model.
For Li/LFP, the theoretical capacities values at C/10 and C/2 are 156 mAh.g-1 and
149 mAh.g-1, respectively, and the corresponding experimental capacity values are 156
mAh.g-1 and 148 mAh.g-1 (Figure 5.1a)). Similar agreement is observed for the Li/LMO
half-cell, with theoretical and experimental values of 86 mAh.g-1 and 56 mAh.g-1 at
C/10 and C/2, respectively (figure 5.1b)).
In the higher capacity region (magnification in figure 5.1a) of the discharge curves
there are small deviations between the theoretical simulations and the experimental
results, attributed to corresponding differences in the electronic conductivity values
[29]. Further, the theoretical electronic conductivity values described by equation (2) do
not take into account the microscopic physico-chemical phenomena associated to
electrical resistance that occurs on carbon black particles dispersed together with the
active material. Further, the voltage difference between theoretical and experimental
values is higher at the C/2 discharge rate than at the C/10 discharge rate, which is
associated to internal total resistance effects at high discharge rates [30].
In any case, a good theoretical approximation is obtained for both discharge curves
(C/10 e C/2) and half-cells, allowing the validation of the theoretical model.
0 15 30 45 60 75 903.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2 b)
C10
C2
Voltage
/ V
vs L
i/Li+
Capacity / mAh.g-1
5. Theoretical simulation of the cathode
98
5.4.2 Influence of the cathode components content in the performance of the half-
cell.
The effect of the content of the different components of the cathode in Li/LFP and
Li/LMO half-cells performance was first evaluated by taking into account batteries with
different active material (C1) content. Then, for each battery with a specific C1, C2 and
C3 were varied in order to obtain discharge curves at a discharge rate of 1C, as shown in
figure 5.2a) and 5.2b) for Li/LFP and figure 5.3b) for Li/LMO half-cells, respectively.
Figure 5.2 - Voltage as a function of delivered capacity for Li/LFP half-cells with C1:
95% a) and 50% b) at a discharge rate of 1C.
Figure 5.2a) shows a representative Li/LFP half-cell with 95% of C1 with C3
varying from 0.7% to 1,8%. When C3 varies, C2 changes accordingly. It is observed that
when C3 is below 1% there is instability on battery operation and losses in the capacity
value, whereas when C1 is above 1%, a constant capacity value is obtained. These
results show that there is a minimum value of C3 to maintaining the battery with low
internal resistance (see impedance values later in section 5.4.3) and without capacity
losses. It is important to notice that the minimum value of C3 also depends on the active
material content, as shown in figure 5.2. Further, the C3 content only affects the
electrical conductivity, the porosity value of the cathode remaining constant at a value
of ε=71%. The porosity of the cathode just varies ~ 1% with varying C3 for a specific C1
content. Thus, the porosity is more affected by C1, due to the higher density of the
material.
Figure 5.2b) shows the Li/LFP half-cell with C1=50% and C3 ranging from 8% to
40%. The behaviour of this half-cell is representative of the other ones with different
0 20 40 60 80 100 120 140 160 180
2.4
2.6
2.8
3.0
3.2
3.4
a)
Volta
ge
/ V
vs L
i/L
i+
Capacity / mAh.g-1
C3
0.7%
0.8%
0.9%
1.0%
1.5%
1.8%
0 10 20 30 40 50 60 70 80
2.4
2.6
2.8
3.0
3.2
3.4
b)
Volta
ge
/ V
vs L
i/L
i+
Capacity / mAh.g-1
C3
8%
8.6%
9.6%
10%
12%
40%
5. Theoretical simulation of the cathode
99
active material contents. It is important to notice that with decreasing C3 there is also a
decrease in the value of the capacity, indicative of poor stability in battery operation.
For C3 ranging from 8% to 40% the obtained values of the capacity range from 9.3
mAh.g-1 to 81.1 mAh.g-1. It is to notice that in this case, the minimum value of C3 is
higher than for the half-cell with 95% of C1. For the half-cell with a C1= 50%, the
minimum C3 is 10%. This effect is explained by the balance between the increase of the
capacity associated to the higher active material content, and the losses associated to the
internal resistance: lower active material content implies a lower ionic current that
should be compensated by an electronic conduction to maintain a high performance
battery.
Figure 5.3a) and 5.3b) show ten batteries for each half-cell (Li/LFP and Li/LMO)
with C1 contents ranging 50% to 95%. In each of these batteries the C3 and
consequently the C2 was varied.
Figure 5.3 - Delivered capacity as a function of C3 for different C1 for Li/LFP (a) and
Li/LMO (b) half-cells at a discharge rate of 1C.
0 5 10 15 20 25 30 35 40 45 50
0
20
40
60
80
100
120
140
160
180a)
C1=50%
C1=55%
C1=60%
C1=65%
C1=70%
C1=75%
C1=80%
C1=85%
C1=90%
C1=95%
Ca
pa
city / m
Ah
.g-1
C3 / %
0 5 10 15 20 25 30 35 40 45 50
0
10
20
30
40
50 b)
C1=50%
C1=55%
C1=60%
C1=65%
C1=70%
C1=75%
C1=80%
C1=85%
C1=90%
C1=95%
Ca
pa
city / m
Ah.g
-1
C3 / %
5. Theoretical simulation of the cathode
100
Figures 5.3a) and 5.3b) also show that for batteries with 95% of C1 it is possible to
vary C3 between 0.7% and 4%, thus allowing a minimum of 1% for C2, similar to the
minimum of 2% for C2 content that has been reported experimentally [14].
It is observed that Li/LFP and Li/LMO half-cells show a different minimum C3,
above which a constant capacity value is obtained, leading to higher battery
performance. High conductive material content increases the electrical conductivity, but
does not contribute to an increase of the capacity of the battery, once the amount of
active material (quantity of ions) limits the capacity value, as observed for the two
active materials (figure 5.3). For low active material contents, high percentages of
conductive material are required, once it is necessary to optimize the electrical
conduction to obtain maximum capacity values of the battery, as a low electrical
conductivity implies a low profitability of the intercalation of ions within the cathode
along the discharge cycle.
Figure 5.4 - Minimum percentage of C3 as a function of C1 for both half-cells at a
discharge rate of 1C.
Figure 5.4 shows a summary of minimum percentage of C3 supporting an optimal
battery performance for both half-cells with different amount of C1. By decreasing the
active material content from 95% to 50%, the minimum percentage of C3 increases
linearly. The same behaviour is obtained for both LFP and LMO based batteries.
Table 5.2 shows the ratio (n) between C2 and C3 (equation 8) for both half-cells at
1C.
50 60 70 80 90 1000
2
4
6
8
10 Li/LFP
Li/LMO
Co
nd
uctive
mate
rial conte
nt, C
3 / %
C1 / %
5. Theoretical simulation of the cathode
101
Table 5.2 - Minimum values of n=C2/C3 as a function of C1 for the Li/LFP and Li/LMO
half-cells at a discharge rate of 1C.
Half-Cell C1
Li/LFP 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
n (C2/C3) 40/10 36/9 32/8 28/7 24/6 20/5 16/4 12/3 8/2 4/1
Li/LMO 50% 55% 60% 65% 70% 75% 80% 85% 90% 95%
n (C2/C3) 40/10 36/9 32/8 28/7 24/6 20/5 16/4 12/3 8/2 4/1
Table 5.2 shows that a constant value of the n is obtained (n= 4) for the different
active material contents for both half-cells. Thus, this ratio is independent of the nature
and of the type of active material used for half-cell fabrication. The ratio n depends of
the electrical conductivity value for neat conductive material that was used in battery.
In summary, in order to obtain an optimal half-cell performance, a minimum of C3
of 25% has to be used in relation to the C2. This minimum ratio is validated by using
carbon black as conductive material [14].
Figure 5.5 compares the simulated capacity values (Capacitysim) with the theoretical
capacity values (Capacitytheo) for the Li/LFP half-cell with different active material
contents at 1C discharge rate. Independently of the active material content it is observed
a good agreement with small differences between the theoretical and experimental
results. The observed differences are attributed to the effect of charge-transfer resistance
and the electronic/ionic conductivity value assumed in the theoretical model. Capacitysim
were obtained at the minimum ratio n where the battery operates with better stability.
Figure 5.5 - Delivered capacity and Capacitysim/Capacitytheo (%) ratio as a function of
C1 for the Li/LFP half-cell at 1C discharge rate.
50 60 70 80 90 10060
80
100
120
140
160
180
De
live
red
ca
pa
city / m
Ah
.g-1
C1 / %
40
60
80
100
Ca
pa
city
sim
/Ca
pa
city
the
o/ %
5. Theoretical simulation of the cathode
102
Figure 5.5 shows that for C1=50% the decrease of the capacity is also approximately
50% of the theoretical capacity. Therefore, C1 contents above 50% should be selected
for suitable half-cell performance.
Further, the performance of the half-cell was evaluated at low, medium and high
discharges rates (1C, 5C and 10C) (Figure 5.6a) and 5.6b)).
Figure 5.6 - a) Delivered capacity as a function of minimum C3 for the Li/LFP half-
cells: a) C1=95% at 1C, 5C and 10C discharge rates and b) C1 = 95%, 75% and 50% at
5C discharge rate.
As mentioned before, the minimum of C3 to obtain a stable performance in a battery
with C1=95% is 1% at 1C, being obtained (figure 5.6) 1.1% and 1.3% at 5C, 10C
discharges rates, respectively (figure 5.6a)). Thus, the high ionic conductivity required
for high discharges rates is obtained for electrical conductivity. Thus, it is possible to
conclude that once achieved the electrical percolation network for C1 = 95%, the
minimum value of C3 is similar for discharge rate.
Figure 5.6b) also shows that increasing C1 leads to a decrease of the minimum C3 to
maintaining a stable battery. Further, the minimum C3 also decreases with decreasing
discharge rate (see figure 5.3 and 5.6b)).
Table 5.3 shows the minimum values of n obtained for the Li/LFP half-cells with
different C1 at 1C, 5C and 10C discharge rates.
0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0
0
40
80
120
160
1C
5C
10C
De
live
red
ca
pa
city / m
Ah
.g-1
C3 / %
a)
0 2 4 6 8 10 12 40 45 50
0
20
40
60
80
100
120
C1=50%
C1=75%
C1=95%
De
livere
d c
ap
acity / m
Ah.g
-1
C3 / %
b)
5. Theoretical simulation of the cathode
103
Table 5.3 - Minimum values of the n ratio for different C1 for Li/LFP half-cells at 1C,
5C and 10C discharge rates.
C1
Discharge Rate 95% 75% 50%
1C n=(4/1) = 4 n=(20/5) = 4 n=(40/10) = 4
5C n=(3.9/1.1) = 3.54 n=(19.5/5.5) = 3.54 n=(39/11) = 3.54
10C n=(3.7/1.3) = 2.64 n=(18.5/6.5) = 2.86 n=(37/13) = 2.84
Table 5.3 shows that n increases with increasing discharge rate for a given active
material content and that for a fixed discharge rate, the variable n is independent of the
active material content.
Thus, the minimum C3 depends on C1 and discharge rate value. However, the ratio n
is independent of C1, but depends on the discharge rate. At high discharges rates, it is
required high ionic and electrical conduction to obtain a facilitated intercalation process.
So, it is important to take into account the minimum value of n according to the battery
operation discharge rate.
5.4.3 Impedance of the LFP and LMO half-cells
The impedance of the half-cells was evaluated through the Nyquist plots to better
understand the conduction phenomena according to the balance of the different cathode
components in the Li/LFP and Li/LMO half-cells. Based on the previous sections, the
battery resistance was evaluated with the minimum C3 and ratio n for different C1, as
obtained in the previous study at a discharge rate of 1C.
For the different half-cell simulations, the Nyquist plots are characterized by a
semicircle at high frequencies (the overall resistance) and an approximately 45º line in
the low-frequency range, which can be considered as the Warburg impedance,
associated with the lithium-ion diffusion in the bulk of the active material [31].
Figure 5.7a) and 5.7b) shows the Nyquist plot with C1 of 95% and 50% at 1C
discharge rate, respectively. Figure 5.7a) shows a total impedance that corresponds to
the sum of the electrolyte resistance (Re, high frequency intercept with the Z´-axis),
surface film resistance (Rf, Li-ion migration resistance through the solid electrolyte
interface (SEI) film formed on the cathode surface) and charge-transfer reaction
resistance (Rct) ascribed to the lithium-intercalation process. It is observed that an
increase of the impedance value is observed below 1.0% of C3, which explains the
5. Theoretical simulation of the cathode
104
a)
b)
existence of a minimum C3 in order to maintain the battery operating with high
performance and stability at a given discharge rate. At a discharge rate of 1C and a C3
below 1%, the capacity value decreases significantly for the Li/LFP half-cell with a C1
of 95%. When the battery is characterized by a high resistance, the normal intercalation
process of the cathode along the discharge cycle is affected. The same behaviour is
observed for the Li/LFP half-cell with C1=50% at 1C (figure 5.7b). The total impedance
values for C3 = 9.8%, 10% and 12%, are 0.0034 .m2, 0.0030 .m2 and 0.0010 .m2,
respectively. The minimum value of C3 is 10%, as shown in figure 5.4a), the resistance
of the battery increases for lower C3 contents. Figure 5.7c) shows the Nyquist plot for
two Li/LMO half-cells. The total impedance value of Li/LMO with C1 = 95% and C3 =
1% being 0.0076 .m2 and for Li/LMO half-cells with C1 = 50% and C3 =10% the real
impedance value is 0.016 .m2. Thus, despite the half-cell with C1 = 50% showing a
higher C3, the battery resistance value is higher.
4x10-3
5x10-3
5x10-3
6x10-3
6x10-3
7x10-3
7x10-3
8x10-3
0
2x10-4
4x10-4
6x10-4
8x10-4
1x10-3
1x10-3
- Z
'' /
.m2
Z' / .m2
C3
0.9%
1%
2%
3%
4%
4x10-3
5x10-3
6x10-3
7x10-3
8x10-3
9x10-3
0
2x10-4
4x10-4
6x10-4
8x10-4
1x10-3
1x10-3
1x10-3
2x10-3
- Z
'' /
.m2
Z' / .m2
C3
9.8%
10%
12%
40%
5. Theoretical simulation of the cathode
105
c)
Figure 5.7 - Nyquist plot for the Li/LFP half-cell: a) C1 = 95% with different C3 values
at 1C discharge rate and b) C1 = 50% with different C3 values at 1C discharge rate.
Nyquist plot for Li/LMO half-cells: c) C1 = 95% and 50% and C3 = 1% and 10% at 1C
discharge rate.
Figure 5.8 shows the overall impedance values for the Li/LFP and Li/LMO half-
cells obtained for the minimum C3 at different C1 and 1C discharge rate, where a
considerable increase of the real impedance value below the minimum C3 for a given C1
in both batteries is observed.
0 5 10 15 20 25 30 35 40 45 50
0
1x10-3
2x10-3
3x10-3
4x10-3
a)
Z' /
.m
2
C3 / %
C1=95%
C1=90%
C1=85%
C1=80%
C1=75%
C1=70%
C1=65%
C1=60%
C1=55%
C1=50%
2x10-2
3x10-2
5x10-2
6x10-2
0
2x10-2
4x10-2
6x10-2
-Z''
/
.m2
Z' / .m2
C1=95%, C
3=1%
C1=50%, C
3=10%
5. Theoretical simulation of the cathode
106
Figure 5.8 - Total impedance as a function of minimum C3 for different C1 at 1C
discharge rate for: a) Li/LFP and b) Li/LMO half-cells.
For both Li/LFP and Li/LMO half-cells it is observed that above a minimum C3, the
real impedance value remains constant, allowing to optimize the conductive additive
content.
5.4.4 Electrolyte and Electrode Current Density for LFP half-cells
The previous sections showed that there is a minimum value of C3 for a fixed C1 in
order to maintain a good operation of the battery. Now, it is important to qualitatively
evaluate the electrolyte and electrode current density in the cathode to investigate the
behaviour of ions and electrons during the intercalation process at a given time. For the
evaluation of the electrolyte and electrode current density it was chosen the time of 500
s, as this time is within the discharging time range for all evaluated batteries.
The electrolyte current density is defined by the current density of charges
associated to lithium ions that exist in the electrolyte present in the pores of the cathode.
The electrode current density is the current density of charges associated to electrons
moving on the solid phase of the cathode. During the discharge process, the ions move
from the separator towards the current collector, through the empty spaces within the
cathode (pores). At the same time, the electrons move in the opposite direction, from the
current collector to the separator, through the solid phase of the cathode (active and
0 5 10 15 20 25 30 35 40 45 50
0
5x10-3
1x10-2
2x10-2
2x10-2
3x10-2
3x10-2
4x10-2
b)
Z' /
.m2
C3 / %
C1=95%
C1=90%
C1=85%
C1=80%
C1=75%
C1=70%
C1=65%
C1=60%
C1=55%
C1=50%
5. Theoretical simulation of the cathode
107
conductive materials). Figure 5.9 shows a schematic representation of the intercalation
process (reduction of lithium ions) within the cathode during the discharge process.
Figure 5.9 - Schematic representation of a battery cathode and the corresponding
intercalation process during the discharge mechanism.
In figure 5.9, the boundary between the separator and the cathode is located at x= 0
m and the interface of the cathode with the current collector is located at x= 70 m.
Figure 5.10 shows both the electrolyte and the electrode current density along the
width of the cathode at the time of 500 s for a Li/LFP half-cell with 95% of active
material and 4% of conductive material at 1C discharge rate. As a control parameter, it
is shown that the conservation of charge is respected
5. Theoretical simulation of the cathode
108
0 1x10-5
2x10-5
3x10-5
4x10-5
5x10-5
6x10-5
7x10-5
0,0
1,5
3,0
4,5
6,0
7,5
9,0
Cu
rre
nt D
en
sity / A
.m2
Cathode length / m
Electrolyte
Electrode
Sum of electrolyte and electrode
Figure 5.10 - Electrolyte and electrode current density as a function of cathode length
for a Li/LFP half-cell with C1 = 95% and C3 = 4% of at 1C discharge rate and at 500s.
The blue line corresponds the sum of both current densities along the width of the
cathode, showing that the divergence of the total electric charge is null.
At t = 500 s, the current density of the electrolyte decreases from x= 0 μm to x= 70
μm, showing that the amount of available lithium ions decreases along of width of the
cathode due to the intercalation process. The electrolyte current density close to the
current collector is lower than it is at the separator, as ions are subjected to the
intercalation process in positions closer to the separator. During the intercalation of
ions, they are neutralized or reduced will quickly decreasing the ionic current through
the electrode.
The results are shown for the Li/LFP half-cells, being also representative for the
Li/LMO half-cell.
Figures 5.11 and 5.12 show the electrolyte and electrode current density along the
width of the cathode at a time of 500 s. The simulations were performed in Li/LFP half-
cells with C1 = 95% and 50% with various C3 at 1C discharge rate.
5. Theoretical simulation of the cathode
109
Figure 5.11 - Electrolyte current density as a function of the cathode length for Li/LFP
half-cell for various C3 at 1C discharge rate and at 500s for C1= 95% (a) and 50% (b).
Regarding the electrolyte current density value for the half-cell with C1 =95% in the
middle position of the cathode at 500 s, it is observed that this value is high for
conductive material contents below 1% of minimum C3, as shown in figure 5.11a).
Also, for the half-cell with C1 = 50% and C3 below 10% there is a significant increase
of the electrolyte current density at the middle position of the cathode, as shown in
figure 5.11b). This phenomenon indicates that below a minimum C3, the intercalation
process of ions occur with deeper magnitude in locations closer to the current collector.
For low conductive material content, the higher electrical resistance within the solid
phase of the cathode, leads to lower electrode current density in locations closer to the
separator/cathode interface. Thus, the electrode current density value is higher positions
0,00 2,50x10-5
5,00x10-5
7,50x10-5
0
2
4
6
8
10
a)
Ele
ctr
oly
te C
urr
en
t D
ensity / A
.m2
Cathode length / m
C3
0.9%
1%
2%
3%
4%
0,00 2,50x10-5
5,00x10-5
7,50x10-5
0
2
4
6
8
10
b)
Ele
ctr
oly
te C
urr
en
t D
en
sity / A
.m2
Cathode length / m
C3
9.8%
10%
12%
40%
5. Theoretical simulation of the cathode
110
close to the cathode/current collector interface, leading to a higher intercalation process
rate in these regions, as shown in figure 5.12a) and 5.12b).
0,00 2,50x10-5
5,00x10-5
7,50x10-5
0
2
4
6
8 a)
Ele
ctr
od
e C
urr
en
t D
en
sity / A
.m2
Cathode lenght / m
C3
0.9%
1%
2%
3%
4%
0,00 2,50x10-5
5,00x10-5
7,50x10-5
0
2
4
6
8b)
Ele
ctr
ode
Curr
ent D
ensity / A
.m2
Cathode lenght / m
C3
9.8%
10%
12%
40%
Figure 5.12 - Electrode current density as a function of cathode length for Li/LFP half-
cell for various C3 at 1C discharge rate and 500s for C1 = 95% (a) and 50% (b).
Figure 5.12 shows that above a minimum C3 for both half-cells with C1 = 95% and
50% the electrode current density values are low in the middle position of the cathode,
as shown the figures 5.12a) and 5.12b), due to the high electrical resistance of the
cathode.
Figure 5.13 shows the electrolyte and electrode current density as a function of time
at 20 µm of position inside of cathode in relation to cathode/separator interface. The
cathode contains C1 = 95% and C3 = 0.9%. The width of cathode is 70 µm.
0 200 400 600 800 1000 1200 1400 1600
7,4
7,6
7,8
8,0
8,2
Ele
ctr
oly
te C
urr
en
t D
en
sity /
A.m
-2
Time / s
0,6
0,8
1,0
1,2
1,4
Ele
ctro
de C
urre
nt D
ensity
/ A.m
-2
Figure 5.13 - Electrolyte and electrode current density as a function of time for a
Li/LFP half-cell with C1 = 95% and C3 = 0.9% at 20 µm of position inside of cathode in
relation to separator/cathode interface. The width of the cathode is 70 µm.
5. Theoretical simulation of the cathode
111
It is observed that the electrolyte current density increases for the first 600 s and
decreases for larger times. The behavior observed for the electrode current density is
the opposite of the one observed for the electrolyte current density. The electrolyte and
electrode current densities are symmetric to each other. After 600 s, the electrolyte
current density decreases, the reason for this fact is due that in this position begins to
occur with more intensity the intercalation of the ions taking into account that its
density decreases resulting one increase of the electrical resistance.
The behavior of electrolyte and electrode current density as a function of time is
independent of the cathode position.
For low conductive material content, the electrons are subjected to higher resistance
in their flux, so the lithium ions move deeper inside the cathode before the intercalation
process occurs. Thus, at one instant of time of discharge the half-cells with less
conductivity material content show the higher electrolyte density current and lower
electrode current density in the middle position of cathode.
5.5 Conclusions
The optimization of the electrode formulation based on different active material
content, binder and conductive additive is essential for maximizing the electrode
properties in lithium-ion batteries. Thus, this work reports on the optimization of the
electrode formulation for two active materials: C-LiFePO4 and LiMn2O4. The
theoretical simulations were based on the Doyle/Fuller/Newman theoretical model and
the validation of the theoretical model was performed through comparison with
experimental results.
It was found that the C2/C3 ratio described by the variable n should be taken into
account in the fabrication of the cathode. Independently of the active material type, the
minimum value of the C2/C3 ratio is 4 at a discharge rate of 1C. So, when the battery is
subjected to a discharge rate of 1C, the relationship C3 = 0.25×C2 should be respected.
The minimum value of the C2/C3 ratio depends on the discharge rate, as well as the
electrical conductivity, which depends on the C2/C3 ratio and the electrical conductivity
value of neat conductive material.
The ideal relation for the electrode material is 90% of percentage of active material
(C1) for obtain good cycling and the C2 and C3 varying between 2 and 8% according the
scan rate and respecting the mechanical stability.
5. Theoretical simulation of the cathode
112
5.6 References
1. Evarts, E.C., Lithium batteries: To the limits of lithium. Nature, 2015.
526(7575): p. S93-S95.
2. Tarascon, J.M. and M. Armand, Issues and challenges facing rechargeable
lithium batteries. Nature, 2001. 414(6861): p. 359-367.
3. Vincent, C.A., Lithium batteries: a 50-year perspective, 1959–2009. Solid State
Ionics, 2000. 134(1–2): p. 159-167.
4. Whittingham, M.S., Lithium Batteries and Cathode Materials. Chemical
Reviews, 2004. 104(10): p. 4271-4302.
5. Daniel, C. and J.O. Besenhard, Handbook of Battery Materials2012: Wiley.
6. Winter, M. and R.J. Brodd, What Are Batteries, Fuel Cells, and
Supercapacitors? Chemical Reviews, 2004. 104(10): p. 4245-4270.
7. Wakihara, M. and O. Yamamoto, Lithium Ion Batteries: Fundamentals and
Performance2008: Wiley.
8. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
9. Fergus, J.W., Recent developments in cathode materials for lithium ion
batteries. Journal of Power Sources, 2010. 195(4): p. 939-954.
10. Newman, J., Optimization of Porosity and Thickness of a Battery Electrode by
Means of a Reaction‐Zone Model. Journal of The Electrochemical Society,
1995. 142(1): p. 97-101.
11. Ramadesigan, V., et al., Optimal Porosity Distribution for Minimized Ohmic
Drop across a Porous Electrode. Journal of The Electrochemical Society, 2010.
157(12): p. A1328-A1334.
12. Zheng, H., et al., Calendering effects on the physical and electrochemical
properties of Li[Ni1/3Mn1/3Co1/3]O2 cathode. Journal of Power Sources, 2012.
208: p. 52-57.
13. Zheng, H., et al., Cooperation between Active Material, Polymeric Binder and
Conductive Carbon Additive in Lithium Ion Battery Cathode. The Journal of
Physical Chemistry C, 2012. 116(7): p. 4875-4882.
5. Theoretical simulation of the cathode
113
14. Gören, A., et al., State of the art and open questions on cathode preparation
based on carbon coated lithium iron phosphate. Composites Part B:
Engineering, 2015. 83: p. 333-345.
15. Martínez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
16. Chen, Y.-H., et al., Selection of Conductive Additives in Li-Ion Battery
Cathodes: A Numerical Study. Journal of The Electrochemical Society, 2007.
154(10): p. A978-A986.
17. Sousa, R.E., C.M. Costa, and S. Lanceros-Méndez, Advances and Future
Challenges in Printed Batteries. ChemSusChem, 2015. 8(21): p. 3539-3555.
18. Miranda, D., C.M. Costa, and S. Lanceros-Mendez, Lithium ion rechargeable
batteries: State of the art and future needs of microscopic theoretical models
and simulations. Journal of Electroanalytical Chemistry, 2015. 739: p. 97-110.
19. Ram, P., et al., Improved performance of rare earth doped LiMn2O4 cathodes
for lithium-ion battery applications New Journal of Chemistry, 2016. 40: p.
6244-6252.
20. Wang, S., L. Lu, and X. Liu, A simulation on safety of LiFePO4/C cell using
electrochemical–thermal coupling model. Journal of Power Sources, 2013. 244:
p. 101-108.
21. Safari, M. and C. Delacourt, Modeling of a Commercial Graphite/LiFePO4 Cell.
Journal of The Electrochemical Society, 2011. 158(5): p. A562-A571.
22. Yu, S., et al., Model Prediction and Experiments for the Electrode Design
Optimization of LiFePO4/Graphite Electrodes in High Capacity Lithium-ion
Batteries. Bulletin of the Korean Chemical Society, 2013. 34(1): p. 9.
23. Wang, M., et al., The effect of local current density on electrode design for
lithium-ion batteries. Journal of Power Sources, 2012. 207: p. 127-133.
24. Srinivasan, V. and J. Newman, Discharge Model for the Lithium Iron-Phosphate
Electrode. Journal of The Electrochemical Society, 2004. 151(10): p. A1517-
A1529.
25. Dai, Y., L. Cai, and R.E. White, Simulation and analysis of stress in a Li-ion
battery with a blended LiMn2O4 and LiNi0.8Co0.15Al0.05O2 cathode. Journal
of Power Sources, 2014. 247: p. 365-376.
5. Theoretical simulation of the cathode
114
26. Lestriez, B., Functions of polymers in composite electrodes of lithium ion
batteries. Comptes Rendus Chimie, 2010. 13(11): p. 1341-1350.
27. Young, R.J. and P.A. Lovell, Introduction to Polymers, Third Edition2011:
Taylor & Francis.
28. Van Zee, J.W., et al., Advances in Mathematical Modeling and Simulation of
Electrochemical Processes and Oxygen Depolarized Cathodes and Activated
Cathodes for Chlor-alkali and Chlorate Processes1998: Electrochemical
Society.
29. Wang, C. and J. Hong, Ionic/Electronic Conducting Characteristics of LiFePO4
Cathode Materials: The Determining Factors for High Rate Performance.
Electrochemical and Solid-State Letters, 2007. 10(3): p. A65-A69.
30. Ning, G., B. Haran, and B.N. Popov, Capacity fade study of lithium-ion batteries
cycled at high discharge rates. Journal of Power Sources, 2003. 117(1–2): p.
160-169.
31. Shi, Y., et al., Graphene wrapped LiFePO4/C composites as cathode materials
for Li-ion batteries with enhanced rate capability. Journal of Materials
Chemistry, 2012. 22(32): p. 16465-16470.
6. Computer simulations of 2D interdigitated batteries
115
6. Computer simulation evaluation of the geometrical
parameters affecting the performance of two
dimensional interdigitated batteries
This chapter describes the simulation of the effect of the geometrical parameters of
interdigitated batteries, including the number, thickness and the length of the digits, on
the delivered battery capacity. This optimization was carried out in two dimensions
maintaining the area of the different components constant.
This chapter is based on the following publication:
“Computer simulation evaluation of the geometrical parameters affecting the
performance of two dimensional interdigitated batteries”, D. Miranda, C. M. Costa, A.
M. Almeida, S. Lanceros-Méndez, Journal of Electroanalytical Chemistry 781 (2016) 1-
11.
6. Computer simulations of 2D interdigitated batteries
117
6.1 Introduction
Lithium-ion batteries are nowadays the most relevant and efficient energy storage
systems, increasingly used for applications in portable electronic products, such as
mobile-phones, computers, e-labels and disposable medical testers, hybrid electric
vehicles (HEVs) and electric vehicles (EVs) [1].
The rechargeable battery market is expected to reach $ 22.5 billion dollars and the
growth of the lithium- ion battery market in 2016 is expected to reach 25%. The
increasing demands of the automotive and mobile phone sectors result in an increasing
need for lithium ion battery autonomy, power and capacity [2].
The widespread presence of lithium-ion batteries is due to their advantages in
comparison with other battery systems, as they are lighter, cheaper, have higher energy
density (between 100 and 265 Wh kg-1), lower self-discharge, no memory effect,
prolonged service-life and higher number of charge/discharge cycles [3, 4]. Improving
lithium-ion battery performance is nevertheless needed with respect to specific energy,
power, safety and reliability [4].
Typically, the performance of a battery is optimized for either power or energy
density by modifying the chemistry and materials for electrodes (anode and cathode)
and separators in conventional two-dimensional structures [5-7]. This structure is
defined as a layer-by-layer configuration such as cathode/separator/anode [8].
Nevertheless, this structure is limited by the slow transport of lithium ions and
hindered accessibility to the material at the back of the electrode, close to the current
collector [9].
Taking this limitation into account and in order to maximize power and energy
density, interdigitated structures are being developed [9]. The interdigitated geometry
consists of electrode arrays of rods separated by a solid electrolyte, i.e, lithium salts put
directly into the polymeric matrix without organic solvent present in electrolyte. In this
way, the surface area of the electrodes increases without additional side reactions on the
electrode surfaces [10].
This configuration leads to shorter Li+ transport paths, reducing ion diffusion
lengths and electrical resistance across the entire battery system, as well as to higher
energy density of the cell within the same areal footprint [9-11].
6. Computer simulations of 2D interdigitated batteries
118
In this context, interdigitated batteries using high capacity manganese oxide
cathodes and lithium anode have achieved a capacity of up to 29.5 μAh/cm2, which is
10x the average capacity of rechargeable conventional batteries [12].
Three dimensional (3D) interdigitated architectures have been fabricated by
printing concentrated LFP-LTO based inks, showing a high areal energy density of 9.7 J
cm-2 at a power density of 2.7 mW cm -2 [13].
3D printing was also used for the fabrication of batteries based on Li4Ti5O12
(average particle diameter of 50 nm) and LiFePO4 (average particle diameter of 180
nm). This battery (960 μm × 800 μm, electrode width = 60 μm, spacing = 50 μm) shows
a high areal energy density of 9.7 J cm-2 at a power density of 2.7 mWcm-2 [13].
The interdigitated architecture mostly depends on the aspect ratios (length/width)
that can be achieved as well as on the geometry of the electrode. In this way, computer
simulations of battery performance are important and critical for evaluating the
optimized geometries before experimental implementation [14, 15].
In order to simulate battery operation, the couplings of different physical-chemical
levels are needed. Macroscopic models allow geometrical and dimensional optimization
of the battery components and mesoscale models are suitable for understanding and
improving the different components of the battery: physical-chemical properties of the
materials to be used as electrodes and separators and the choice of the most suitable
organic solvents for electrolytes [15-17]. Theoretical simulations on 3D battery
architectures have been addressed by focusing on determining an optimal electrode
cylinder array configuration [18] as well as the planar tessellated electrode geometry of
square and circular electrode arrays, in which the cell capacity can be increased by
simply adding more electrodes in the plane of the array or increasing the height of the
electrodes [19]. Further, a Finite Element Analysis (FEA) electrochemical model has
been developed for several of the main 3D battery architectures such as interdigitated
cylinders, concentric cylinders and interdigitated plates using a non-porous electrode
(particle-scale) electrochemistry model [20]. The effect of the solid electrolytes ionic
conductivity was also analyzed for interdigitated structures, the discharge capacity
increasing with increasing of ionic conductivity [21, 22].
Theoretical simulation was also used to demonstrate that the electrode thickness
can significantly influence many key aspects of a battery such as energy density,
temperature response, capacity fading rate and overall heat generation, among others
[23].
6. Computer simulations of 2D interdigitated batteries
119
In 3D pillar structures, template pillar heights (h) and interpillar distances (d) have
been evaluated, the optimum pillar height being ~ 70 µm in order to achieve
homogeneous lithiation and high cell capacity [24].
The influence of geometry in the performance of conventional and unconventional
lithium-ion batteries was studied maintaining the same area of the different components
and it has been shown that the geometry with the best performance is the interdigitated
structure [25].
Taking into account the state-of-the art on 3D battery architecture simulation and
that interdigitated structures maximize the performance of the battery, the goal of this
work is focus in the quantitative evaluation of the effect of the variation of the
geometrical parameters of the interdigitated structure towards performance optimization
of lithium-ion batteries. The considered geometrical parameters are the number,
thickness and length of the digits, and the optimization has been performed considering
different scan rates. To our knowledge these effects have never been comprehensively
reported before and it is important to take them into account before experimentally
implementing the adequate geometry of a battery for particular applications, allowing to
improve battery design for specific area restrictions. The performance of the battery was
determined in two dimensions at different scan rates up to 400C, as the combination of
interdigitated structure fabrication with printing technologies allows to obtain
interdigitated batteries with small size and thickness and yet with high delivered
capacity. The optimization of the interdigitated structure by a FEA was carried out
taking into account the number, thickness and length of the digits, while maintaining the
area of the different components constant. The results are also compared with a
conventional structure. As a result, optimization of the geometrical parameters of
interdigitated geometries is achieved, allowing to guide experimental fabrication by
providing an essential tool for proper battery design and implementation.
6.2 Theoretical simulation model and parameters
The main components of lithium ion batteries are anode, cathode and separator, that
can be simulated by the Doyle/Fuller/Newman model in two dimensions (2D) [26]. The
electrochemical model used is presented in Chapter 3.
The nomenclature and the physical meaning of the different symbols are shown in
6. Computer simulations of 2D interdigitated batteries
120
the List of Symbols and Abbreviations.
In this work, a finite element method is implemented, considering the
electrochemical and transport processes in interdigitated lithium ion battery structure
such as: [porous positive electrode, (LixMn2O4) | porous separator, poly(vinylidene-
trifluoroethylene) (P(VDF-TrFE)) soaked in 1M lithium hexafluorophosphate (LiPF6) in
propylene carbonate (PC) | porous negative electrode, (LixC6)], the simulations being
performed in 2D. The degree of porosity of the electrodes is defined as the space
between the particles of active electrode material and the respective values are shown in
Table 6.1.
Figure 6.1 represents a conventional (figure 6.1a)) and an interdigitated (figure
6.1b)) geometry with the identification of the investigated geometrical parameters in
three (3D) and two dimensions (2D).
Figure 6.1 - Schematic representation of a conventional (a) and an interdigitated (b)
battery with indication of the main geometrical parameters.
6. Computer simulations of 2D interdigitated batteries
121
In this case, the control of the active mass loading in both electrodes is achieved
through the volume of the electrodes. As the study was performed in 2D, the mass
loading is related to the area.
Figure 6.1a) shows a 3D interdigitated battery in which the volume of each
electrode corresponds to the multiplication of the dimension L by the area of the
electrode, Aa and Ac for the anode and the cathode, respectively. For the 2D model, on
the other hand, the mass loading is just related with the area of each electrode (figure
6.1b). Here, the cathode area is larger than the anode area (Ac > Aa) on all conventional
and interdigitated batteries as shown by the values assigned to each electrode (Table
6.1).
The volume of active material for a 3D geometry is determined by the active
material content through its initial concentration (initial parameter indicated in table 6.1,
CE,i,0).
The same principle was applied for the interdigitated geometry (figure 6.1b)).
In order to study the influence of geometrical parameters (number of digits, length
and thickness of the digits) in the discharge capacity value at a specific scan-rate, it is
necessary to maintain the same area of each component whenever a specific parameter
is changed.
Thus, the same active mass loading of both electrodes is maintained, as well as the
degree of porosity in the electrolyte and separator, allowing to maintain constant the
capacity and just to evaluate the effect of the geometrical parameters.
Figure 6.2 illustrates how the area is maintained for the various battery components
(electrodes, separator and current collectors) when varying the number of digits, from 4
to 2, of an interdigitated battery.
6. Computer simulations of 2D interdigitated batteries
122
Figure 6.2 - Schematic representation illustrating how the area of each component is
maintained constant, while varying the number of digits.
When the number of digits decreases, the area of the active material (mass of the
active material) that was interdigitated will be moved to part of the electrode which is
not interdigitated.
This fact is illustrated in figure 6.2b) by Aaa and Acc1 for the anode and the cathode,
respectively. In this way, the mass of active material remains constant independently of
the variation of the geometrical parameters.
The values of the parameters used for each component of the battery are listed in
Table 6.1. In the computer simulations, the length of the digit (c_dig), the thickness
(e_dig) and the number were varied while maintaining constant the area of both
electrodes (Aa and Ac), separator (As) and current collectors (Acc).
Relatively to the parameters of the separator, the constant values are indicated in
Table 6.1 and the variable parameter is its thickness (e_sep).
6. Computer simulations of 2D interdigitated batteries
123
Table 6.1 - Parameters used in the simulations of the conventional and interdigitated
battery structures.
Parameters used for the simulation of both conventional and interdigitated structures
Parameter Unit Anode (LixC6) Separator Cathode (LixMn2O4)
CE,i,0 mol/m3 14870 3900
CE,i,max mol/m3 26390 22860
CL mol/m3 1000
r m 12,510-6 810-6
Kl S/m 6,510-1 6,510-1 6,510-1
Keff,i S/m (6,510-1) 0,3571,5 (6,510-1) 0,4441,5
Kf S/m (6,510-1) (4,8410-2)
Dl m2/s 4,010-10 4,010-10 4,010-10
Deff,i m2/s (4,010-10) 0,3571,5 (4,010-10) 4,8410-2 (4,010-10) 0,4441,5
t0+ 0,363 0,363 0,363
DLI m2/s 3,910-14 110-13
Brugg or p 1,5 8,5 1,5
f,i 0,172 0,259
i 0,357 0,70 0,444
3,8
i S/m 100 3,8
i1C
A/m2 17,5
F C/mol 96487
T K 298,15
R J/mol K 8,314
Ai m2 4,010-8 1,810-9 8,010-8
Geometrical parameters used for the conventional structure
Parameter Unit Anode (LixC6) Separator Cathode (LixMn2O4)
Li m 20010-6 9010-6 40010-6
Geometrical parameters used for the interdigitated structure
Parameter Unit Anode (LixC6) Separator Cathode (LixMn2O4)
c_dig m c_dig c_dig
e_dig m e_dig e_dig
e_sep m e_sep
N 1 to 8 1 to 8
The finite element calculations were carried out using a MATLAB subroutine in
order to solve the governing equations of the constituents (electrodes and separator) in
an ideal cell without SEI formation. The size of the mesh is one order of magnitude
below the dimension of the components.
The value of C-rate was determined from the cathode electrode area taking into
account the corresponding active material.
The impedance was measured for each geometry at frequencies ranging from 10
mHz to 1 MHz with a potential perturbation with an amplitude of 0.01 V and with the
following parameters: film resistance of the positive electrode: 0.0065 m2.S-1; film
resistance of the negative electrode: 1×10-5 m2.S-1; double layer capacitance of the
positive electrode: 0.2 F.m-2; double layer capacitance of the negative electrode: 0.2
F.m-2; current collector resistance at each current collector: 1.1×10-4 m2.S-1.
6. Computer simulations of 2D interdigitated batteries
124
6.3 Results
Theoretical model simulations of the lithium-ion battery were applied for studying
the influence of different geometrical parameters, including the number of digits (N),
their length (c_dig) and thickness (e_dig), in the interdigitated geometry (figure 6.1b))
and the results were compared to those obtained for a conventional structure at low,
medium and high discharge rates. In all simulations, the area of the different
components, anode, cathode and separator, was maintained constant in order to keep the
same amount of active material and to evaluate only the effect of the geometrical
differences. Further, the same area was used for the current collectors in all simulations,
in order to maintain the same ohmic resistance. The capacity value is in the form of
ampere-hour per square meter (Ah.m-2) – capacity per unit area depending on the
electrode area for optimizing the geometrical parameters.
6.3.1 Conventional geometry
Figure 6.1a) shows the schematic representation of a battery with a conventional
geometry (conventional battery). Figure 6.1a) also shows the geometrical variables
which are evaluated at various discharge rates in order to investigate their influence in
the capacity of the battery: thickness of the anode, La, thickness of the cathode, Lc, and
thickness of the separator, e_sep.
It is important to notice that for the conventional structure, increasing the thickness
of each component implies to increase the area of the battery, once the height of the
battery is constant.
The choice of the initial dimensions for the anode and the cathode is related to the
fact that the amount of active material for the cathode should be higher in comparison to
the active material for the anode. In the discharge process ions move from the anode to
the cathode, the active mass loading of the anode working as lithium ions source and,
therefore, the higher the mass loading (area in this case) of the anode, the higher will be
the capacity value in the discharge process taking into account the area of the cathode
and respecting the cell balance.
6. Computer simulations of 2D interdigitated batteries
125
The intercalation process of ions occurs in the cathode during the discharge
process, the area of the cathode being larger in order to increase the number of
intercalation ions in this process.
Further, it is also considered that the areas of the conventional and interdigitated
geometries are maintained constant [27], in order to allow proper comparison between
both battery types and to properly consider the effect of the variation of the geometrical
parameters (number, thickness and length of digit) in the interdigitated geometry.
Figure 6.3a) shows the delivered capacity measured at 1C discharge rate as a
function of the anode thickness with the cathode and separator widths fixed at 400 μm
and 25 μm, respectively. The thickness of the anode was varied from 200 μm to 540
μm, with a step of 20 μm, the cathode and initial anode areas are 8×10-8 and 4×10-8 m2,
respectively.
200 250 300 350 400 450 500 550
700
800
900
1000
1100
1200 a)
De
livere
d c
ap
acity / A
h.m
-2
Thickness of anode / m
150 200 250 300 350 400 450
400
450
500
550
600
650
700
750
800
b)
De
live
red
ca
pa
city / A
h.m
-2
Thickness of cathode / m
Figure 6.3 - Delivered capacity at 1C discharge rate as a function of the anode thickness
for a fixed cathode thickness of 400 μm (a) and as a function of the cathode thickness
for a fixed anode thickness of 200 μm (b).
Figure 6.3a) shows that varying the thickness of the anode from 200 μm to 400 μm
leads to an increase of the capacity value from 750 Ah.m-2 to 1207 Ah.m-2, reaching a
constant value for the anode thickness above 400 μm.
Figure 6.3b) shows the influence of the variation of the thickness of the cathode
(140 to 420µm) for a fixed anode (200µm) and separator thickness (25µm) in that the
initial cathode and anode areas are 2.8×10-8 and 4×10-8 m2, respectively.
It is observed that varying the thickness of the cathode between 140 μm and 260
μm leads to increased battery capacity values from 422 Ah.m-2 to 750 Ah.m-2 and that
6. Computer simulations of 2D interdigitated batteries
126
for cathode thickness larger than 260 μm the capacity of the battery remains constant.
The thickness of the cathode is higher in comparison to the thickness of the anode due
to the possibility to obtain larger variations of the discharge rates, as presented in figure
6.4.
Taking into account that the ideal value of the cathode thickness is 400 μm (Figure
6.3b)), Figure 6.4 shows the delivered capacity for different anode thicknesses and a
constant separator thickness of 90 μm. This separator thickness value allows a simpler
variation of the geometrical parameters for the interdigitated geometry, maintaining the
areas of the components constant (electrodes and separator) in both geometries
(interdigitated and conventional).
0 50 100 150 200 250 300 350 400
0
200
400
600
800
1000
1200
0 25 50 75 100400
600
800
1000
1200
Deliv
ere
d c
apacity / A
h.m
-2
Scan rate / C
200 m
300 m
400 m
De
livere
d c
ap
acity / A
h.m
-2
Scan rate / C
200 m
300 m
400 m
Figure 6.4 - Delivered capacity as a function of the scan rate for three different anode
thicknesses and fixed cathode thickness of 400 μm.
Figure 6.4 shows that a conventional geometry with a separator thickness of 90 μm
and both electrodes with an equal dimension of 400 μm does not work for discharge
rates above 50C.
So, it is important to reduce the thickness of the anode to obtain a battery which
operates properly at low, medium and high discharge rates. By decreasing the anode
thickness to 300 μm, the battery operates up to a maximum discharge rate of 300C with
a capacity value of 20.25 Ah.m-2.
6. Computer simulations of 2D interdigitated batteries
127
When decreasing the thickness of the anode to 200 μm, the battery operates up to a
maximum discharge rate of 350C, thereby increasing the discharge rate range.
Thus, the delivered capacity depends on the scan rate and on the thickness of the
electrodes, as shown in Figure 6.4. These effects are larger at higher discharge rates, in
which an elevated ionic flow between the electrodes is required, i.e., higher ion
insertion capacity in the cathode.
6.3.2 Interdigitated geometry
For the interdigitated geometry, it was evaluated the influence of the geometrical
parameters (number of digits, width and thickness) on battery capacity at low, medium
and high discharge rates. The results were compared with the ones obtained for the
conventional geometry (section 6.3.1).
The values of the areas chosen for each battery component are presented in Table
6.1. For the selection of the areas it was taken into account the need of having large
areas to allow a wide variation range in the number of digits, keeping the digit thickness
and length constant.
Figure 6.1b show the schematic representation of the simulated interdigitated
geometry in which number of digits (N), digit length, c_dig, and digit thickness, e_dig,
are represented.
6.3.2.1 Influence of the number of digits at different scan rates
Figures 6.5a) and 6.5.b) show the delivered capacity at scan rates from 1C to 400C
for a conventional battery structure and an interdigitated structure with 1 to 8 digits with
a digit thickness of 20 μm and a digit length of 100 μm. It is observed that a constant
capacity is obtained in the range of discharge rates from 1C to 10C for all battery
geometries.
This effect is due to the fact that the discharge rates are quite low, allowing the
mobility of lithium ions from the anode to the cathode and the full insertion of lithium
ions in the cathode. For discharge rates from 50C to 400C, the delivered capacity of the
interdigitated geometry is higher than the capacity of the battery with a conventional
geometry, the capacity value being related with the increase of the number of digits. The
6. Computer simulations of 2D interdigitated batteries
128
main difference between both geometries is the path of ions and electrons, i.e, the ohmic
resistance [10] and the increased contact surface area of the electrodes, leading to
improvement of the insertion of lithium ions in the cathode.
0 50 100 150 200 250 300 350 400
0
150
300
450
600
750
900
1 2 3 4 5 6 7 8 9 10720
725
730
735
740
745
750
755
Deliv
ere
d c
apacity /A
h/m
2
C-rate
1 digit
2 digit
3 digit
4 digit
5 digit
6 digit
7 digit
8 digit
a)
De
live
red c
ap
acity / A
h.m
-2
C-rate
Conventional
geometry
0 1 2 3 4 5 6 7 8 9
0
100
200
300
400
500
600
700
800b)
De
livery
cap
acity / A
h.m
-2
Digit number
C-rate
1
3
5
10
50
150
200
300
350
400
0 50 100 150 200 250 300 350 400
0
100
200
300
400
500
600
700
800c)
De
live
red
ca
pa
city
/ A
h.m
-2
C-rate
c_dig=100m:
e_dig=20m
e_dig=50m
0 1 2 3 4 5 6 7 80
20
40
60
80
100
Se
pa
rato
r th
ickn
ess /
m
Digit number
200
400
600
800
1000
1200
1400d)
Ba
tte
ry w
idth
/
m
Figure 6.5 - Delivered capacity as a function of the scan rate (a and c) and number of
digits (b). Separator thickness and battery width as a function of the number of digits
with a fixed c_dig at 400 μm and e_dig at 20 μm (d).
The conventional geometry (Figure 6.5a)) shows a delivered capacity of 0.7 Ah.m-2
for 350C whereas for the interdigitated geometries this value is much higher, being
49.68 Ahm-2 for of the battery with two digits and 323.77 Ah.m-2 for the battery with 8
digits.
On the other hand, the maximum scan rate and the delivered capacity value are
lower for the interdigitated battery with one digit when compared with the conventional
battery (figure 6.5a)). This behavior is ascribed to the increased length of the
interdigitated battery with one digit, maintaining the same area for both geometries.
6. Computer simulations of 2D interdigitated batteries
129
Figure 6.5b) shows that the delivered capacity remains constant around 737 Ah.m-2
for both the interdigitated and the conventional geometries up to 50C discharge rate. On
the other hand, for scan rates from 50C to 400C, it is verified an increase of the capacity
value from the conventional geometry to the interdigitated geometry with eight digits.
The delivered capacity for the interdigitated geometry with two different digit
thicknesses (e_dig= 20 m and 50 m) for equal digit length (c_dig=100 m) shows
that the larger thickness improves the delivered capacity of the interdigitated geometry,
independently of the scan rate (Figure 6.5c)). This effect is due to the fact that the
interdigitated geometry with larger digit thickness (e_dig) leads to shorter battery width
between the electrodes, as the area of the components is maintained.
Figure 6.5d) shows that the length of the battery for the conventional geometry is
about 700 m and for the interdigitated geometry with one digit is 1300 m. Although
the thickness of the separator has been decreased from 90 m to 60 m in the
conventional geometry and the interdigitated geometry with one digit, respectively, the
charges should move through longer pathways, leading to higher ohmic losses.
For further analysing the effect of the different geometries, the electrochemical
impedance spectra was investigated evaluate the mass transport phenomena during the
discharge of the battery [28]. Figure 6.6 shows the typical impedance curve (Nyquist
plot) for the conventional and the interdigitated geometry with 8 digits.
0 5x10-5
1x10-4
1x10-4
2x10-4
3x10-4
3x10-4
4x10-4
0
1x10-4
2x10-4
3x10-4
4x10-4
5x10-4
6x10-4
b a
-Z''
/
.m2
Z' / .m2
Figure 6.6 - Nyquist plot for the conventional (a) and the interdigitated (b) geometry
with 8 digits in frequency range of 1 mHz to 1MHz.
6. Computer simulations of 2D interdigitated batteries
130
Independently of the geometries, each plot in Figure 6.6 is characterized by two
semicircles at high frequencies, representing the ohmic resistance, ionic resistance due to
the pores and interfacial charge-transfer resistance. The inclined line in the low-
frequency range of Figure 6.6 corresponds to the Warburg impedance, associated with
the lithium-ion diffusion in the bulk of the active material [29]. The total impedance
represented by the diameter of the semicircles is observed to be higher for the
conventional geometry than for the interdigitated geometry.
6.3.2.2 Influence of length and thickness of the digit
In the previous section (6.3.2.1) it was observed that the length and thickness of
digit of the interdigitated geometry affect more significantly the delivered capacity at
higher scan rates. This effect will be analysed in detail in the following sections for a
battery with four digits operating at a discharge rate of 400C. The study of the influence
of digit length and digit thickness variation in the delivered capacity is performed for
separators either with constant or variable width. The effect of the geometrical
parameters is evaluated for a fixed separator thickness, whereas in some cases it is
necessary to modify the thickness of the separator in order to keep constant the area of
the different components.
6.3.2.2.1 Influence of digit length from 60 μm to 480 μm
Figure 6.7 shows the influence of the digit length for a constant digit thickness of
20 m in the delivered capacity of a four digits battery (Figure 6.7a)) and the
corresponding effects in the width of the battery (Figure 6.7b)) for both constant and
variable separators. For the interdigitated geometry with constant separator, the
separator thickness value is 16.49 m, which corresponds to a digit length of 100 m
and a digit thickness of 20 m.
6. Computer simulations of 2D interdigitated batteries
131
50 100 150 200 250 300 350 400 450 500
200
210
220
230
240
250
260
II
a)
De
live
red
ca
pa
city / A
h.m
-2
Digit length / m
I
4 6 8 10 12 14 16 18 20 22Separator thickness / m
50 100 150 200 250 300 350 400 450 500
480
520
560
600
640
680
720II
I
b)
Digit length / m
Battery
wid
th /
m
Figure 6.7 - a) Delivered capacity and b) width of the battery as a function of digit
length for a four digits battery for a constant (I) and a variable (II) separator.
Independently of the separator type, Figure 6.7a) shows that the delivered capacity
increases with increasing digit length due to the increased contact surface between the
electrodes and therefore to the decrease of the ion pathways. For digit lengths between
60 m and 100 m, the delivered capacity for the variable separator is higher in
comparison with the interdigitated geometry with a constant separator.
For digit lengths larger than 100 m, the interdigitated geometry with a constant
separator shows higher delivered capacity (Figure 6.7a)). The reason for this fact is
observed in Figure 6.7b) and depends essentiality on the width of the battery. Figure
6.7a) also shows the variation of the delivered capacity as a function of the separator
thickness for a variable separator, the delivered capacity decreasing with increasing
separator thickness.
It is also observed in Figure 6.7b) that the width of the battery with a constant
separator is larger in comparison with the variable separator up to a digit length of 100
m. The area of the electrodes is thus constant and the thickness of the separator
decreases, which implies an increase in the length of the battery to maintain the same
area of the electrodes.
This fact is also supported by the impedance curves for the three digit lengths
shown in Figure 6.8, which shows that the total impedance of the semicircles decreases
as the digit length increases.
6. Computer simulations of 2D interdigitated batteries
132
0 5x10-5
1x10-4
1x10-4
2x10-4
3x10-4
3x10-4
4x10-4
0
1x10-4
2x10-4
3x10-4
4x10-4
5x10-4
6x10-4
70 m
150 m
480 m
-Z''
/
.m2
Z' / .m2
Figure 6.8 - Nyquist plot of interdigitated geometries for three different digit lengths in
the frequency range from 1 mHz to 1MHz.
.
6.3.2.2.2 Influence of the digit thickness from 10 μm to 70 μm
Figure 6.9 shows the influence of the digit thickness in the delivered capacity
(Figure 6.9a)) and width of the battery (Figure 6.9b)) for both constant and variable
separators and with a constant digit length of 100 m. For the interdigitated geometry
with constant separator, the separator thickness value is 16,49 m, corresponding to a
digit length of 100 m and a digit thickness of 20 m.
10 20 30 40 50 60 7050
100
150
200
250
300
350
II
a)
De
livere
d c
ap
acity / A
h.m
-2
Digit thickness / m
I
12 14 16 18Separator thickness / m
10 20 30 40 50 60 70
200
300
400
500
600
700
800
II
Ib)
Digit thickness / m
Ba
tte
ry w
idth
/
m
Figure 6.9 - a) Delivered capacity and b) width of the battery as a function of digit
thickness for a constant (I) and a variable (II) separator.
6. Computer simulations of 2D interdigitated batteries
133
Independently of the separator type, Figure 6.9a) shows an increase of the delivered
capacity with increasing digit thickness. This effect is due to the increasing contact
surface area between each electrode and the separator and the reduction of the width of
the battery (Figure 6.9b)). Taking into account the areas of each of the components, the
digit thickness increase was limited to 70 μm.
The increase of the delivered capacity as a function of digit thickness is related to
the decrease of the battery width for both separator types. Figure 6.9a) also shows that
the delivered capacity decreases with increasing separator thickness for the variable
separator. Figure 6.10 shows the Nyquist plot for three digits thickness with a constant
digit length of 100 m and four digits, showing that the total impedance decreases with
increasing digit thickness, affecting the discharge value of the battery.
0 7x10-5
1x10-4
2x10-4
3x10-4
3x10-4
4x10-4
0
1x10-4
2x10-4
3x10-4
4x10-4
5x10-4
6x10-4
70m 30m10m
-Z''
/
.m2
Z' / .m2
Figure 6.10 - Nyquist plot of the interdigitated geometries for three different digit
thicknesses in the frequency range from 1 mHz to 1MHz.
6.3.2.2.3 Maximum limit for digit thickness and length at 200C and 400C
The influence of the maximum limit values for thickness and length in the delivered
capacity was evaluated. These so called “digit limits” are the maximum possible values
maintaining constant the area of the interdigitated geometry illustrated in the figure
6.11.
6. Computer simulations of 2D interdigitated batteries
134
Figure 6.11 - Schematic representation of the: a) digit limit length and b) digit limit
thickness for four digits.
The same procedure was carried out for the other interdigitated batteries with
different numbers of digits. It is to notice that the maximum length and width of the
digit value that can be achieved decreases with increasing number of digits.
Figures 6.12a) and 6.12b) show the delivered capacity for batteries with different
number of digits as a function of “digit limit” thickness and length with c_dig= 100m
and e_dig=20m, respectively, for 200C and 400C.
0 50 100 150 200 250 300
240
280
320
360
400
440
480
520a)
400C
200C 1 digit
7 digit
8 digit
6 digit
5 digit
4 digit
3 digit
De
live
red
ca
pa
city / A
h.m
-2
Digit limit thickness / m
2 digit
200 300 400 500 600 700 800 900 10001100
0
75
150
225
300
375
450
525
2 digit
b)
400C
1 digit
7 digit
8 digit 6 digit5 digit 4 digit
3 digit
De
live
red
ca
pa
city / A
h.m
-2
Digit limit length / m
200C
6. Computer simulations of 2D interdigitated batteries
135
1 2 3 4 5 6 7 8
200
400
600
800
1000
1200
1400
1600
c_dig=100m
e_dig=20m c)
Battery
wid
th /
m
Digit number
Figure 6.12 - Delivered capacity as a function of digit limit thickness (a) and length (b)
at 200C and 400C. c) Width of the battery as a function of the number of digits for
c_dig= 100 m and e_dig=20 m at 200C and 400C.
It is observed that the delivered capacity of the battery increases with increasing
number of digits for both scan rates (figure 6.12a) and 6.12b)). The digit limit thickness
(Figure 6.12a)) and length (Figure 6.12b)) decrease with increasing the number of
digits, as it implies a decrease of the maximum digit length and thickness (due to the
fact that a constant area is maintained), leading to a decrease of the battery width (figure
6.12c)).
Figure 6.12a) also shows that the maximum delivered capacity as a function of digit
thickness for 200C is 433 Ah.m-2 for 1 digit and 484.42 Ah.m-2 for 8 digits, which is
related to the increase of the contact surface area between the electrodes and the
separator and the decrease of the thickness of the separator, the width of the battery
being practically constant (Figure 6.12c)). Similarly, the maximum delivered capacity
for 1 digit is 0.2 Ah.m-2 and 481.42 Ah.m-2 for 8 digits (Figure 6.12b)).
Figures 6.11a) and 6.11b) show that the interdigitated battery with 2 digits and digit
limit thickness showed larger capacity when compared with the same battery with digit
limit length. The delivered capacity value for the interdigitated battery with 2 digits and
digit limit thickness is 469 Ah.m-2 and for the corresponding battery with digit limit
length is 300 Ah.m-2. This fact is due to a higher contribution of the width of battery to
the delivered capacity when compared to the contact surface area between the electrodes
and the separator.
6. Computer simulations of 2D interdigitated batteries
136
Finally, Figure 6.12c) shows that the width of the interdigitated battery with 2 digits
and length digit limit is 1400 m, being just 210 m for the same battery with the
thickness digit limit.
6.4. Discussion
Different geometrical parameters have been evaluated for both conventional and
interdigitated geometries in order to optimize the performance of the later one.
For the conventional geometry, the effect of the variation of the anode thickness is
the increase of the battery capacity due to the increasing active material content (lithium
ion content) [30], but Figure 6.4 shows that the limit of lithium ions at the cathode is
reached: during the discharge process, the cathode receives lithium ions coming from
the anode, but due to its thickness, there is a maximum capacity of insertion of these
ions.
Thus, the choice of the anode and cathode dimension is fundamental in the
conventional geometry in order to obtain a high delivered capacity, i.e, the cathode
thickness should be equal or higher than the anode thickness for the investigated
electrochemical system.
Figures 6.5a)-6.5d) show that the interdigitated geometry shows higher delivered
capacity in comparison to the conventional geometry, as the former geometry facilitates
the mobility of ions between electrodes. At medium and high scan rates a fast mobility
of the ions is required, resulting in higher charge flow for both electrons and ions and a
larger ion insertion ability in the cathode. In a conventional battery, the mobility of ions
is hindered by the larger paths that ions must travel from the anode to the cathode
(larger width of the battery), as well as by the larger thickness of the separator (higher
resistance to ionic conductivity). Also the charge-transfer resistance value affects the
battery performance, as can be seen through the impedance curves (Figure 6.6).
Taking into account the results shown in Figures 6.5 to 6.12 for the interdigitated
geometry, it is concluded that paths for ions between the electrodes is substantially
reduced, the contact surface between the electrodes is improved and the thickness of the
separator is reduced in comparison to conventional geometry, while maintaining the
area of the different components constant.
6. Computer simulations of 2D interdigitated batteries
137
The thickness and the length of the digits are relevant parameters as they are related
to the lithium-ion insertion in the cathode material during the discharge process and
smaller ion paths lead to the observed variations in the delivered capacity. The ohmic
losses related to the width of the battery can be reduced by increasing the contact
surface area, resulting in an increase of the delivery capacity in the interdigitated
geometry.
6.5. Conclusions
Interdigitated structures are essential for obtaining maximum power and energy
density in battery systems. In this way, the optimization of the geometrical parameters
such as the number, thickness and length of the digits is required for optimizing battery
performance, while maintaining constant the area of the different components. This
optimization was performed in this work in two dimensional interdigitated structures,
following the Doyle/Fuller/Newman theoretical model.
With respect to the geometry optimization of the interdigitated geometry, it was
observed that increasing the number of digits implies an increase in the capacity of the
battery due to the smaller path of the lithium ions between electrodes in the
intercalation/deintercalation process.
For the same digit number, increasing the thickness and the length of the digits
leads to an increase in the capacity of the battery as the width of the battery decreases,
leading to reduced ohmic losses associated to charge transport and increased surface
contact area of the electrodes, which facilitates the insertion process on the cathode
material during the discharge process.
The interdigitated geometry increases the contact surface area between each
electrode and the separator and thereby increases the corresponding ion flow. Thus, it is
concluded that, if maintaining the same areas for all components, the interdigitated
geometry strongly improves the delivered capacity value in comparison to the
conventional geometry, this improvement being particularly relevant at high discharge
rates.
6. Computer simulations of 2D interdigitated batteries
138
6.6 References
1. Wakihara, M. and O. Yamamoto, Lithium ion batteries: fundamentals and
performance1998: Kodansha.
2. Gonzalez, F. and P. Harrop, Batteries & Supercapacitors in Consumer
Electronics 2013-2023: Forecasts, Opportunities, Innovation, 2014, IDTechEx.
3. Whittingham, M.S., Lithium Batteries and Cathode Materials. Chemical
Reviews, 2004. 104(10): p. 4271-4302.
4. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
5. Goriparti, S., et al., Review on recent progress of nanostructured anode materials
for Li-ion batteries. Journal of Power Sources, 2014. 257(0): p. 421-443.
6. Chikkannanavar, S.B., D.M. Bernardi, and L. Liu, A review of blended cathode
materials for use in Li-ion batteries. Journal of Power Sources, 2014. 248(0): p.
91-100.
7. Lee, H., et al., A review of recent developments in membrane separators for
rechargeable lithium-ion batteries. Energy & Environmental Science, 2014.
7(12): p. 3857-3886.
8. Long, J.W., et al., Three-Dimensional Battery Architectures. Chemical Reviews,
2004. 104(10): p. 4463-4492.
9. Rohan, J.F., et al., Energy Storage: Battery Materials and Architectures at the
Nanoscale. ICT - Energy - Concepts Towards Zero - Power Information and
Communication Technology2014.
10. Arthur, T.S., et al., Three-dimensional electrodes and battery architectures. MRS
Bulletin, 2011. 36(07): p. 523-531.
11. Pikul, J.H., et al., High-power lithium ion microbatteries from interdigitated
three-dimensional bicontinuous nanoporous electrodes. Nat Commun, 2013. 4:
p. 1732.
12. Pikul, J.H., P.V. Braun, and W.P. King, High power primary lithium ion
microbatteries. Journal of Physics: Conference Series, 2013. 476(1): p. 012087.
13. Sun, K., et al., 3D Printing of Interdigitated Li-Ion Microbattery Architectures.
Advanced Materials, 2013. 25(33): p. 4539-4543.
6. Computer simulations of 2D interdigitated batteries
139
14. Ramadesigan, V., et al., Modeling and Simulation of Lithium-Ion Batteries from
a Systems Engineering Perspective. Journal of The Electrochemical Society,
2012. 159(3): p. R31-R45.
15. Miranda, D., C.M. Costa, and S. Lanceros-Mendez, Lithium ion rechargeable
batteries: State of the art and future needs of microscopic theoretical models and
simulations. Journal of Electroanalytical Chemistry, 2015. 739(0): p. 97-110.
16. Martínez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
17. Franco, A.A., Multiscale modelling and numerical simulation of rechargeable
lithium ion batteries: concepts, methods and challenges. RSC Advances, 2013.
3(32): p. 13027-13058.
18. Hart, R.W., et al., 3-D Microbatteries. Electrochemistry Communications, 2003.
5(2): p. 120-123.
19. Liang, R.H.P., et al., Mathematical modeling and reliability analysis of a 3D Li-
ion battery. J. Electrochem. Sci. Eng., 2014. 4(1): p. 17.
20. Zadin, V., et al., Modelling electrode material utilization in the trench model
3D-microbattery by finite element analysis. Journal of Power Sources, 2010.
195(18): p. 6218-6224.
21. Itoh, F., G. Inoue, and M. Kawase, Reaction and Mass Transport Simulation of
3-Dimensional All-Solid-State Lithium-Ion Batteries for the Optimum Structural
Design. ECS Transactions, 2015. 69(1): p. 83-90.
22. Zadin, V. and D. Brandell, Modelling polymer electrolytes for 3D-
microbatteries using finite element analysis. Electrochimica Acta, 2011. 57: p.
237-243.
23. Zhao, R., J. Liu, and J. Gu, The effects of electrode thickness on the
electrochemical and thermal characteristics of lithium ion battery. Applied
Energy, 2015. 139: p. 220-229.
24. Priimägi, P., et al., Optimizing the design of 3D-pillar microbatteries using finite
element modelling. Electrochimica Acta, 2016. 209: p. 138-148.
25. Miranda, D., et al., Computer simulations of the influence of geometry in the
performance of conventional and unconventional lithium-ion batteries. Applied
Energy, 2016. 165: p. 318-328.
6. Computer simulations of 2D interdigitated batteries
140
26. Doyle, M., et al., Comparison of Modeling Predictions with Experimental Data
from Plastic Lithium Ion Cells. Journal of The Electrochemical Society, 1996.
143(6): p. 1890-1903.
27. Daniel, C., Materials and processing for lithium-ion batteries. JOM, 2008. 60(9):
p. 43-48.
28. Huang, R.W.J.M., F. Chung, and E.M. Kelder, Impedance Simulation of a Li-
Ion Battery with Porous Electrodes and Spherical Li + Intercalation Particles.
Journal of The Electrochemical Society, 2006. 153(8): p. A1459-A1465.
29. Zhu, J.G., et al., A new electrochemical impedance spectroscopy model of a
high-power lithium-ion battery. RSC Advances, 2014. 4(57): p. 29988-29998.
30. Favors, Z., et al., Stable Cycling of SiO2 Nanotubes as High-Performance
Anodes for Lithium-Ion Batteries. Sci. Rep., 2014. 4.
7. Computer simulations of different battery geometries
141
7. Computer simulations of the influence of geometry in
the performance of conventional and unconventional
lithium-ion batteries
This chapter evaluates the influence of the battery geometry in the performance of
lithium-ion batteries. In order to optimize battery performance, different geometries
have been evaluated taking into account their suitability for different applications. These
different geometries include conventional and interdigitated batteries, as well as
unconventional geometries such as horseshoe, spiral, ring, antenna and gear batteries.
This chapter is based on the following publication:
“Computer simulations of the influence of geometry in the performance of conventional
and unconventional lithium-ion batteries”, D. Miranda, C. M. Costa, A. M. Almeida, S.
Lanceros-Méndez, Applied Energy 165 (2016) 318-328.
7. Computer simulations of different battery geometries
143
7.1 Introduction
Energy storage systems are an essential need in a modern society with rapid
technological advances, increasing mobility and environmental concerns [1-3], the most
used energy storage systems being lithium-ion batteries [4, 5].
Lithium-ion batteries are essential in applications such as mobile-phones and
computers, among others. Further, they area also explored for hybrid electric vehicles
(HEVs) and electric vehicles (EVs) [6-8].
Lithium-ion batteries dominate the battery market with a share of 75% due to their
advantages with respect to other battery systems (NiCd, nickel-cadmium and NiMH,
nickel-metal hydride), including high energy density, lightweight, high average
discharge rate, no memory effect and high cycle life [9, 10].
The key issues for lithium-ion batteries are related to improving specific energy,
power, safety and reliability [5]. These issues strongly depend on the materials for
electrodes (anode and cathode) and separator (porous membrane with electrolyte
solution) [11-14].
Together with the materials, also the geometry of the battery strongly affects its
performance, the interdigitated geometry being the most investigated for this effect [15-
17].
The improving specific energy, power, safety and reliability of lithium ion batteries
are strongly depend on the materials for electrodes (anode and cathode) and separator
(porous membrane with electrolyte solution) [11-14]. Together with the materials, also
the geometry of the battery strongly affects its performance, the interdigitated geometry
being the most investigated for this effect [15-17].
The interdigitated geometry is based on electrode digits separated by an electrolyte,
allowing increased surface area for the electrodes. In this geometry, the Li+ transport
paths are shorter, reducing the electrical resistances across the battery and ion diffusion
[16, 18].
As an example, lithium-ion microbatteries with interdigitated electrodes have been
fabricated by electrodepositing high capacity electrolytic materials, manganese oxide
cathode and lithium anode. The capacity value of these microbatteries is 29.5
μAh/cm2μm, with an increase in capacity and power by 10x and 1000x, respectively, in
comparison with conventional batteries [16, 19].
7. Computer simulations of different battery geometries
144
Microbatteries based on interdigitated geometries have been fabricated by printing
Li4Ti5O12 (LTO) and LiFePO4 (LFP) based inks. These batteries show high energy
density, 9.7 J cm-2, at a power density of 2.7 mW cm-2 and can be used in
microelectronics and biomedical devices [20].
The combination of printing technologies and microbatteries allow to obtain
customizable thin batteries with large area and at low-cost [21]. These batteries can be
fabricated with specific geometries by different printing (screen, spray and inkjet
printing) techniques, depending on the final applications. Thus, it has been
demonstrated that it is possible to fabricate microbatteries by ink-jet printed that operate
at 90 C [22].
Printed battery applications include radio-frequency identification (RFID), security,
thin film medical products and products that require on-board battery power [23]. Thus,
evaluation of the possible battery geometries is necessary for optimizing size,
fabrication and integration before experimental implementation. The optimization of the
geometries can be carried out through computer simulations of battery performance
[24].
Battery performance by computer simulation is based in models at different
physical levels describing the physical-chemical properties of the materials to be used as
electrodes and separators, as well as the operation of the battery [25-27].
These computer simulations are thus essential for battery development as they
allow the correlation between theoretical and experimental results through the
electrochemical behavior of the batteries [28].
The state-of-the art regarding battery geometry optimization of lithium-ion batteries
through simulation models include interdigitated [16, 18, 29, 30], cylindrical [31, 32],
spiral wound [33] and prismatic geometries [34]. For these geometries, thermal analysis
has been performed [32, 34-36]. Further, different active material shapes for the anode,
i.e, different microstructures [37] have been evaluated as well as the effect of thickness
[38]. Further, the effect of lithium distribution and concentration [39] and geometric
characteristics, i.e, porosity and tortuosity [40] have been computer simulated. Finally, a
theoretical analysis of potential and current distributions has been carried out for
lithium-ion batteries with planar electrodes [41].
The most relevant geometry for increasing capacity value is the interdigitated
geometry [19].
7. Computer simulations of different battery geometries
145
Taking into account the advantages of printing techniques allowing battery
fabrication with unconventional geometries, which will improve device integration and
overall performance for different application, the novelty of this work is to
quantitatively evaluate the effects of seven different lithium-ion battery geometries
while maintaining constant the area of the different components. In this way, just the
effect of geometry variation is quantified. Five of the evaluated geometries have never
been reported before. Battery performance has been determined up to 500C, as
microbatteries fabricated by printing batteries are already able to operate at at high scan
rates above 90C. In this way, battery geometry will be able to be tailored for specific
applications.
The optimization of the seven geometries (conventional, interdigitated, horseshoe,
spiral, ring, antenna and gear) was carried out by finite element method simulations
through the Doyle/Fuller/Newmann model. The choice of the different geometries is
based on their applicability in different devices, including smart-phones, watches,
tables, sensors and RFID tags, among others.
7.2 Theoretical simulation model and specific parameters for each geometry
The Doyle/Fuller/Newman model used in this work describes the main equations
that govern the operation of a battery and its main components: anode, cathode and
separator [42]. The equations of the electrochemical model applied in the simulations
are presented in Chapter 3. The nomenclature and definition of the symbols within the
equations are shown in the List of Symbols and Abbreviations.
Considering the electrochemical and transport processes in a typical lithium-ion
battery structure such as: [anode, (LixC6) | electrolyte/separator, porous membrane of
P(VDF-TrFE) soaked in 1M LiTFSi-PC | cathode, (LixMn2O4)], in this work, a finite
element method is implemented through the previous equations (Doyle/Fuller/Newman
model) for the study of the different geometries shown in table 7.1. The choose of the
thickness of separator is based in [43].
The values of the parameters used for the different components of each battery
geometry are listed in Table 7.1. The areas of all components were maintained constant
in the computer simulations. In the different geometries represented in Table 7.1, d_max
and d_cc represent the maximum distance of the ions to the collectors and the distance
7. Computer simulations of different battery geometries
146
between current collectors, respectively. This table also shows the main characteristics
of each geometry as well as some potential applications.
Table 7.1 - Parameters used for the simulations, main characteristics and applications
for the different battery geometries [44-46].
Parameter Unit Anode (LixC6) Separator Cathode (LixMn2O4)
CE,i,0 mol/m3 14870 3900
CE,i,max mol/m3 26390 22860
CL mol/m3 1000
r m 12,510-6 810-6
Li m 20010-6 40010-6
e_sep m 9010-6
kef,i S/m (6,510-1) 0,3571,5 (6,510-1) 4,8410-2 (6,510-1) 0,4441,5
Def,i m2/s (4,010-10) 0,3571,5 (4,010-10) 4,8410-2 (4,010-10) 0,4441,5
DLI m2/s 3,910-14 110-13
Brugg or p 1,5 8,5 1,5
f,i 0,172 0,259
i 0,357 0,70 0,444
3,8
i S/m 100 3,8
i1C
A/m2 17,5
F C/mol 96487
T K 298,15
R J/mol K 8,314
Ai m2 4,010-8 1,810-9 8,010-8
Specific parameters for each battery geometry
Conventional battery geometry
Parameter Value /
m
Characteristics Applications
Lc 400×10-6
- low surface contact
area between
electrodes
- high separator
thickness
- high distance
between current
collectors
- high ohmic losses
- layer by layer
fabrication
Portable
devices and
electric
vehicles
La 200×10-6
e_sep 90×10-6
d_max 697×10-6
d_cc 690×10-6
7. Computer simulations of different battery geometries
147
Interdigitated battery geometry
Parameter Value /
m
Characteristics Applications
N 8* except
unit - high surface contact
area between
electrodes
- medium distance
between current
collectors
- thin thickness of
separator
- digits
Sensors and
actuators, electric
vehicles, smart
cards
c_dig 100×10-6
e_dig 20×10-6
e_sep 8,66×10-6
d_max 391×10-6
d_cc 327×10-6
Horseshoe battery geometry
Parameter Value /
m
Characteristics Applications
Lc 33,1×10-6 - high surface
contact area between
electrodes
- low distance
between current
collectors
- thin thickness of
separator
- large space at the
center
- layer by layer
fabrication in u form
Portable devices
with empty space
for the placement of
electronic
components at the
center of the
battery, such as,
smart-phones,
tablets and portable
computers
La 17,5×10-6
e_sep 7,71×10-6
d_max 1125×10-
6
d_cc 58,3×10-6
Spiral battery geometry
Parameter Value /
m
Characteristics Applications
Lc 28,6×10-6 - high surface
contact area between
electrodes
- medium distance
between current
collectors
- medium thickness
of separator
- low space at the
center
Smart cards, smart
toys, sensors and
actuators
La 17,8×10-6
e_sep 7,27×10-6
d_max 1240×10-
6
d_cc 53,7×10-6
7. Computer simulations of different battery geometries
148
Ring battery geometry
Parameter Value /
m
Characteristics Applications
Lc 27,4×10-6
- high surface
contact area between
electrodes
- low distance
between current
collectors
- thin thickness of
separator
- large space at the
center
Watches, mobile
phones, medical
devices
La 14,5×10-6
e_sep 6,40×10-6
Rd 430×10-6
d_max 1350×10-
6
d_cc 48,4×10-6
Antenna battery geometry
Parameter Value /
m
Characteristics Applications
Lc 25,6×10-6 - high surface
contact area
between
electrodes
- medium distance
between current
collectors
- thin thickness of
the separator
- small space at
the center
Smart toys, gift
cards, medical
devices (e.g.
transdermal drug
delivery (TDD)
systems
La 16,0×10-6
e_sep 5,88×10-6
d_max 1225×10-
6
d_cc 47,5×10-6
Gear battery geometry
Parameter Value /
m
Characteristics Applications
N 8* except
unit
- high surface
contact area
between
electrodes
- small distance
between current
collectors
- thin thickness
of separator
- large space at
the center
- digits
Watches, mobile
phones, medical
devices
e_sep 12,41×10-6
Rg 93,9×10-6
d_max 294×10-6
d_cc 135,8×10-6
e_dig 40×10-6
c_dig 30×10-6
7. Computer simulations of different battery geometries
149
7.3 Results and Discussion
Theoretical model simulations were thus applied for studying seven different
lithium-ion battery geometries by keeping constant the areas of the anode, cathode,
separator and current collectors. The main objective is to evaluate the effect of the
geometry in battery performance. The delivery capacity was obtained for all geometries
at low, medium and high discharge rates.
7.3.1 Effect of battery geometry
For the different geometries, the current collectors are located in the positions
shown in table 7.1. The choice for the specific position of the collectors for each battery
geometry is based on having the same electric field applied to the lithium-ions that are
located in the places further away in relation to the current collector positions.
Figure 7.1 shows the capacity values obtained for the different geometries for scan
rates from 1C up to 500C.
0 100 200 300 400 500
0
100
200
300
400
500
600
700
800
De
live
red
ca
pa
city / A
h.m
-2
Scan rate / C
Geometry type:
Conventional
Interdigitated
Horseshoe
Spiral
Antenna
Ring
Gear
Figure 7.1 – Delivered capacity as a function of scan rate for the different batteries.
For each geometry, increasing scan rate leads to a decrease of the capacity value.
This fact is ascribed to ohmic drop polarization [47]. Figure 7.1 shows that the
7. Computer simulations of different battery geometries
150
conventional geometry does not operate above 330C discharge rate, the capacity value
at 330C being 0.58 Ahm-2.
At high discharges rates (> 300C) it is observed that the interdigitated and
conventional geometries show the highest and the lowest capacity in comparison to the
other geometries.
Figure 7.1 also shows that there is a significant difference in the capacity value
between the conventional geometry and the remainder geometries for discharge rates
above 50C.
At 330C, it is possible to classify the geometries into three groups. The first group
is constituted just by the conventional geometry, with a capacity value of 0.58 Ahm-2.
The second group is constituted by the ring, antenna and spiral geometries, with a range
of capacity values from 149 Ahm-2 up to 182 Ahm-2. Finally, the third group is
constituted by the gear and interdigitated geometries that show capacity values from
289 Ahm-2 up to 318 Ahm-2. The horseshoe geometry is located between the second and
the third groups with a capacity value of 216 Ahm-2. The horseshoe geometry shows a
higher capacity than the spiral, antenna and ring geometries and a lower capacity than
the gear and interdigitated battery geometries.
The reason for the different capacity values is ascribed to variations of the
maximum distance and distance between current collectors in the different geometries,
as shown in figure 7.2.
3x10-4
6x10-4
9x10-4
1x10-3
1x10-3
0
50
100
150
200
250
300
350
a)
Spiral
Antenna
Ring
Horseshoe
Gear
Interdigitated
Conventional
De
live
red
ca
pa
city / A
hm
-2
Maximum distance / m
0 2x10-4
4x10-4
6x10-4
0
50
100
150
200
250
300
350
b)
Spiral
Antenna
Ring
Horseshoe
Gear
Interdigitated
Conventional
De
livere
d c
ap
acity / A
hm
-2
Distance between collectors / m
Figure 7.2 - Delivered capacity for the different geometries as a function of a)
maximum distance and b) distance between collectors.
7. Computer simulations of different battery geometries
151
Figure 7.2a) shows the maximum distance that the ions move between the
electrodes during the discharge process. It is observed that the interdigitated and gear
geometries show lower maximum distance between the electrodes than the other
geometries, as these geometries show shorter paths for ions to move. The maximum
distances for interdigitated and gear geometries are 391 m and 294 m, respectively.
This fact implies decreasing ohmic losses in the discharge process. Thus, high capacity
values are obtained for these batteries at high discharge rates. It is also shown that the
conventional geometry shows a lower maximum distance but large capacity losses due
to the larger thickness of the separator with respect to the other geometries. This effect
is ascribed to the fact that the same area is maintained for all components in the
different geometries. The thickness of the separator for the conventional geometry is 90
m and, therefore, the conventional geometry has an ionic flow that is hindered by the
separator at high discharge rates.
The horseshoe geometry, for example, shows a higher maximum distance (1125
m) than the conventional (697 m) battery, but its capacity is higher at high discharges
rates (C) due to the thinner separator (~ 7,71 m).
Figure 7.2b) shows the distance between collectors for the different geometries and
it is observed that the ring, antenna, spiral and horseshoe batteries present almost the
same distance between collectors, being in the range from 47.5 m to 58.3 m. In this
way, these geometries show a thin separator and therefore an improved ionic flow
through the separator. On the other hand, the ring, antenna and spiral geometries show a
larger amount of charges (ions and electrons) further from the current collector
positions, leading to higher ohmic losses due to increased internal resistance of the
battery. The ohmic losses are more significant for the value of the capacity of the
battery than the separator thickness. In conclusion, this effect is the main reason for the
horseshoe geometry presenting a higher capacity value than ring, spiral and antenna
geometries, despite all four geometries having approximately the same separator
thickness.
Figure 7.2b) shows that the gear and interdigitated geometries present a higher
distance between the collectors and a larger separator thickness than the horseshoe,
spiral, ring and antenna geometries. The thickness of the separator for the gear and
interdigitated geometries are 12.41 and 8.66 m, respectively, and the distances
between the collectors are 135.8 m and 327 m, respectively. However, in the gear
7. Computer simulations of different battery geometries
152
and interdigitated geometries a higher delivery capacity is obtained. The improvement
in the delivery capacity is related to the fact that the lower maximum distance of the
ions from the collector position overcomes the capacity losses due to the larger
thickness of the separator.
Finally, it is also worth noticing that the influence of the geometry on battery
performance is higher when the batteries operate at high discharge rates, as shown in
figure 7.1 and that the performance depends on the combination of different parameters,
including the maximum distance of the ions to the current collector, d_max, the distance
between current collectors, d_cc, and the thickness of separator and electrodes.
7.3.2 Influence of the geometrical parameters in battery performance
The influence of specific geometrical parameters of the different geometries
(horseshoe, ring and gear) in battery performance is shown in this section. The choice of
these geometries is based on the delivered capacity obtained in the previous section as
well as their application possibilities. The interdigitated geometry also shows high
capacity values but its optimization has been already addressed in the literature. All
simulations consider the same area for the different components (anode, cathode,
separator and current collectors) and high discharge rates. In the horseshoes geometry,
the main parameters studied are the dimensions and current collector positions. For the
ring geometry, it was studied the effect of the radius of the ring. Finally, the battery
performance of ring and gear geometries was compared.
7.3.2.1 Effect of battery dimensions and current collector positions in the
horseshoe geometry
Due to its specific geometrical features, it is particularly important to evaluate the
influence of the dimensions of the battery and the position of the current collectors on
battery performance for the horseshoe geometry. In this study, simulations were
performed at high discharges rates (500C), as effects associated to battery geometry are
more clearly observed.
7. Computer simulations of different battery geometries
153
7.3.2.1.1 Current collector positions
Three current collector positions were selected (figure 7.3a)): A, B and C which
correspond to 0 m, 562.5 m and 1125 m distance from position A.
0 20 40 60 80 100
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4 b)
Vo
lta
ge
/ V
Delivered capacity / Ahm-2
Position of cc (500C)
Position A: 0 m
Position B: 562.5 m
Position C: 1125 m
Figure 7.3 - a) Schematic representation of the current collector positions and b)
voltage as a function of the delivered capacity for the different current collector
positions.
Figure 7.3b) shows that the battery with collectors placed in C results in the highest
capacity value in comparison to the other collector positions, due to the lower ohmic
losses associated to the movement of the ions to the current collector positions.
Being constant the dimensions of the battery, the observed differences in delivery
capacity are just ascribed to the maximum distance of the lithium ions to the collectors
(Figure 7.4).
7. Computer simulations of different battery geometries
154
0 200 400 600 800 1000 1200
1.0x10-3
1.2x10-3
1.4x10-3
1.6x10-3
1.8x10-3
2.0x10-3
2.2x10-3
2.4x10-3
Maxim
um
dis
tance
/ m
Current collector position / m
0
20
40
60
80
100
De
livere
d c
ap
acity / A
hm
-2
Figure 7.4 - Delivered capacity as a function of current collector positions and
maximum distance of lithium ions.
Figure 7.4 shows that the maximum distance of the most distant ions for the 3
collector position A, B and C is 2250 m, 1687 m and 1125 m, respectively, a
decrease of the maximum distance of the furthest ions in relation to the collector
position leading to shortest paths for ion transport and therefore to lower ohmic losses.
Further, for the collectors placed at positions B and A, the magnitude of the electric
field applied to the ions located at places far from the electrodes is lower.
When the collectors are placed in the central geometrical position, all ions located
far from the electrodes are at similar distances, leading to smaller paths for ion
movement and a larger magnitude of the electric field.
So, it is concluded that the current collector position strongly affects battery
performance for this geometry.
7.3.2.1.2. Dimensions of the battery
Taking into account the previous results, it is important evaluate the influence of
the dimensions of the horseshoe geometry in the performance of the battery at high
discharges rates (500C). The dimension under consideration, L_dim in Figure 7.5a),
was modified from 50 m to 750 m.
7. Computer simulations of different battery geometries
155
0 100 200 300 400 500 600 700 80080
100
120
140
160
180
200 b)
De
livere
d c
ap
acity / A
hm
-2
L_dim / m
Figure 7.5 - a) Schematic representation of the horseshoe battery dimension, L_dim,
and b) delivered capacity as a function of L_dim.
Figure 7.5b) shows that the capacity value increases from 176 Ahm-2 to 198 Ahm-2
when L_dim increases from 50 μm to 250 μm. For further increase from 250 m to 750
m, the capacity values decrease from 198 Ahm-2 to 92Ahm-2. In this way, the optimum
capacity value is 198 Ahm-2 for a horseshoe dimension of 250 μm. It would be expected
that by decreasing L_dim, the capacity values would increase due to a reduction of the
ohmic losses. In contrast, a decrease of the capacity value is obtained for L_dim from
250 μm to 50 μm. This effect is explained by the balance between the gains in capacity
associated to the reduction of the ohmic losses and the higher electric field applied to
the most distant ions, and the decrease of the capacity associated to the thickness
increase of the electrodes. Other possible reasons for this fact are the increased
thickness of the separator (hindering ionic flow through the separator) and the decrease
of the surface contact area between electrodes (decreasing ion insertion in the cathode).
7. Computer simulations of different battery geometries
156
0 100 200 300 400 500 600 700 800
0
2x10-4
4x10-4
6x10-4
8x10-4
1x10-3
1x10-3
d_cc
L_dim / m
Dis
tance
/ m
d_max
Figure 7.6 - Maximum distance and distance between current collectors as a function of
L_dim for the horseshoe geometry.
Figure 7.6 shows that increasing L_dim between 50 m to 750 m leads to an
increase of the distance of the more distant lithium ions from the collectors, d_max, and
a decrease of the distance between collectors, d_cc.
It is important to notice that the lager distance between collectors, d_cc, is due to
the increased thickness of separator and electrodes, leading to larger paths for electrons
to move.
In the range of L_dim from 250 m to 50 m it is observed a decrease of the
capacity value. Although, the distance of the most distant ions to the collectors, d_max,
is lower, there is a larger impact of the losses on the capacity values. As previously
mentioned, these losses are related to the larger thickness of the separator, low surface
contact area between electrodes and larger paths for the movement of electrons from the
electrodes to the collectors.
7. Computer simulations of different battery geometries
157
7.3.2.2 Influence of the radius in the ring geometry
The influence of the radius of the ring battery in battery performance was
investigated. The radius of the ring geometry is defined by Rd, as illustrated in figure
7.7a).
0 100 200 300 400
40
60
80
100
120
140
160
180
200
De
livere
d c
ap
acity / A
hm
-2
Rd / m
b)
Figure 7.7 - a) Schematic representation of the ring geometry and b) delivered capacity
as a function of the radius, Rd.
The maximum distance of the ions that are located in distant places with respect to
the collector position, d_max, is half the ring perimeter (Figure 7.7a)).
In section 7.3.1., it was observed that the ring geometry belongs to the group of
geometries with medium capacity value, together with the ring, antenna and spiral
geometries. The capacity can nevertheless be optimized by varying the radius.
Figure 7.7b) shows the relationship between the radius of the ring and the capacity
for high discharges rates (500C).
For a radius of the ring from 20 m to 93.9 m it was obtained an increase in the
capacity value from 176 Ahm-2 to 192 Ahm-2, respectively, due to the balance between
maximum distance of ions and thickness of separator as illustrated in figure 7.8.
Further, for a radius from 93.9 m to 430 m a decrease in the performance of the
battery is observed (figure 7.7b)). In this way, an optimum capacity value of 192 Ahm-2
is obtained for a ring battery with a radius of 93.9 m.
7. Computer simulations of different battery geometries
158
0 100 200 300 400 500
0
2x10-4
4x10-4
6x10-4
8x10-4
1x10-3
1x10-3
1x10-3
Ma
xim
um
dis
tan
ce
/ m
Rd / m
5x10-5
1x10-4
2x10-4
2x10-4
Cu
rre
nt
co
lecto
r d
ista
nce
/ m
5x10-6
1x10-5
1x10-5
2x10-5
2x10-5
Th
ickn
ess o
f S
ep
ara
tor
/ m
Figure 7.8 - Maximum distance, distance between current collectors and thickness of
the separator as a function of Rd.
Figure 7.8 shows that, for Rd from 20 m to 430 m, that the thickness of separator
decreases from 23 m to 6.40 m and that the distance between the collectors decreases
from 189.5 m to 48.35 m. In this way, the gain in capacity associated to the
decreasing thickness of the electrodes and separator are lower than the capacity losses
due to the increase of the maximum distances of the ions from the current collector
positions.
As a result, the ring geometry can be optimized for specific applications taking into
account its radius.
7.3.2.3 Comparative performance of ring and gear battery geometries
A comparative study of the capacity of the ring (figure 7.7a)) and gear battery
geometries was performed for a 500C discharge rate. Figure 7.9a) shows the gear
geometry, which is characterized by the presence of digits in both electrodes, each digit
defined by its thickness (e_dig) and length (c_dig). Further, Rg defines the radius of the
gear geometry. The simulated gear shows 8 digits in both electrodes (figure 7.9a)). The
maximum distance of the most distant ions to the collector position (d_max) is the same
for both gear and ring geometries. The maximum distance in these geometries is half the
perimeter.
7. Computer simulations of different battery geometries
159
Figure 7.9 – Schematic representation of the gear geometry.
The comparative effect of Rg variation (93.9 m and 20 m) in both gear and ring
geometries is illustrated in figure 7.10. In both cases, the values of the thickness and the
length of the digit is 40 m and 30 m, respectively.
0 50 100 150 200 250
2.6
2.8
3.0
3.2
3.4
3.6
a)
175 180 185 190 195 200 205 210 215
2.6
Vo
lta
ge
/ V
Delivered capacity / Ahm-2
Voltage
/ V
Delivered capacity / Ahm-2
R=93.9m
Ring
Gear
7. Computer simulations of different battery geometries
160
0 50 100 150 200 250
2.6
2.8
3.0
3.2
3.4
3.6b)
175 180 185 190 195 200 205 210 215
2.6
Volta
ge / V
Delivered capacity / Ahm-2
Vo
lta
ge
/ V
Delivered capacity / Ahm-2
R=20m
Ring
Gear
Figure 7.10 – Voltage as a function of the delivered capacity for the ring and gear
geometries with different Rg: a) 93.9 m and b) 20 m.
Figure 7.10a) shows the capacity values for both geometries and a Rg of 93.9 m.
It is relevant to notice that the gear and ring geometries show the same distance between
collectors, d_cc, but that the gear geometry presents a lower separator thickness due to
the presence of the digits. It would be expected a higher capacity value in the gear
geometry than for the ring geometry, as the digits of the former increase the surface
contact area between the electrodes and decreases the thickness of the separator.
However, it is observed that the value of the capacity for the gear geometry is lower
than for the ring geometry. Both geometries show the same maximum distance of 294.8
m and therefore the same ohmic losses.
On the other hand, when Rg decreases to 20 m, the opposite behavior is observed
with respect to the capacity values (figure 7.10b)): the gear geometry show larger
capacity that the ring geometry.
The larger capacity obtained for the gear geometry is associated to the lower
separator thickness and larger contact surface area between electrodes due to the fact
that the same area was maintained for the different battery components (electrodes,
separator and current collectors).
The larger capacity of the ring geometry for the larger Rg is explained in figure
7.11 in terms of the electrolyte potential and electrolyte current density vectors (black
arrows) in the different regions of the battery at a specific discharge time.
7. Computer simulations of different battery geometries
161
Figure 7.11 - Electrolyte potential and electrolyte current density vectors for a) ring and
b) gear geometries.
The gear geometry shows lower capacity due to lithium ion accumulation in the
vertices of the digits, leading to a higher charge density in these regions and a
heterogeneity in the electric potential that leads to local electric fields (images A in
figure 7.11). Local electric fields mean lower ionic flow between electrodes and the
electrolyte current density shows different orientations instead of the radial orientation
observed for the ring geometry (images B in figures 7.11), that does not show local
accumulations of lithium ions.
Figure 7.12 shows the capacity values of the gear and ring geometries at 500C for a
Rg of 20 m. The ring geometry shows a separator thickness of 42.6 m and the gear
geometry of 23.4 m: the different separator thickness is due to the presence of the
digits in the gear geometry that lead to an increase of the contact surface area between
the electrodes. A study was thus carried out in which the area of the separator is
duplicated from the gear to the ring geometries.
7. Computer simulations of different battery geometries
162
0 50 100 150 200 250
2.6
2.8
3.0
3.2
3.4
3.6
Voltage
/ V
Delivered capacity / Ahm-2
R=20m,
Ring, e_sep=42.6m
Gear, e_sep=23.4m
Figure 7.12 - Voltage as a function of the delivered capacity for the ring and gear
geometries with different separator thickness.
Figure 7.12 show that the gear geometry shows a higher capacity value than the
ring geometry. In contrast with the previous study, the losses associated to the
accumulation of lithium ions in the vertices of the digits is not significant in comparison
with the gains of capacity associated to the decrease of the thickness of the separator
and the increase in contact surface area between the electrodes. The gear geometry
shows an optimal thickness for the separator value that allows a better ionic flow
through the separator.
In conclusion, when there is a need to decrease the radius of a circular battery for
specific applications, it is important to introduce digits in the electrodes (gear
geometry).
7. Computer simulations of different battery geometries
163
7.4 Conclusions
Geometry optimization is essential for maximizing energy density in lithium-ion
batteries. This work reports on the optimization of specific battery geometries, based on
their potential applicability.
Seven geometries were theoretical simulated based on the Doyle/Fuller/Newman
theoretical model, including conventional, interdigitated, horseshoe, spiral, ring,
antenna and gear batteries.
It is shown that, independently of the geometry, high discharge rates require higher
ion insertion capacity on the cathode (high surface contact area between electrodes),
smaller paths for charges to move between collectors and electrodes (reduced ohmic
losses), thin thickness of the separator (improved ionic flow) and optimized current
collector positions to decrease the loss of magnitude of the electric field applied to the
most distant ions.
At 330C, capacity values of conventional, ring, spiral, horseshoe, gear and
interdigitated geometries are 0,58 Ahm-2, 149 Ahm-2, 182 Ahm-2, 216 Ahm-2, 289 Ahm-
2 and 318 Ahm-2, respectively.
It is also shown that battery capacity can be tailored for the different geometries
taking into account geometrical parameters such as maximum distance of the most
distant ions, d_max, distance between of current collectors, d_cc and thickness of
separator and electrodes, once the materials are selected. In this way, new battery
geometries with optimized performance can be fabricated to allow better integration
into specific devices.
7. Computer simulations of different battery geometries
164
7.5 References
1. Tarascon, J.-M., Key challenges in future Li-battery research. Vol. 368. 2010.
3227-3241.
2. Wakihara, M. and O. Yamamoto, Lithium Ion Batteries: Fundamentals and
Performance2008: Wiley.
3. Dunn, B., H. Kamath, and J.-M. Tarascon, Electrical Energy Storage for the
Grid: A Battery of Choices. Science, 2011. 334(6058): p. 928-935.
4. Thackeray, M.M., C. Wolverton, and E.D. Isaacs, Electrical energy storage for
transportation-approaching the limits of, and going beyond, lithium-ion
batteries. Energy & Environmental Science, 2012. 5(7): p. 7854-7863.
5. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
6. Dunn, J.B., et al., The significance of Li-ion batteries in electric vehicle life-
cycle energy and emissions and recycling's role in its reduction. Energy &
Environmental Science, 2015. 8(1): p. 158-168.
7. Zhang, Q., et al., Nanomaterials for energy conversion and storage. Chemical
Society Reviews, 2013. 42(7): p. 3127-3171.
8. Patry, G., et al., Cost modeling of lithium-ion battery cells for automotive
applications. Energy Science & Engineering, 2015. 3(1): p. 71-82.
9. Vincent, C.A., Lithium batteries: a 50-year perspective, 1959–2009. Solid State
Ionics, 2000. 134(1–2): p. 159-167.
10. Park, J.K., Principles and Applications of Lithium Secondary Batteries2012:
Wiley.
11. Roy, P. and S.K. Srivastava, Nanostructured anode materials for lithium ion
batteries. Journal of Materials Chemistry A, 2015. 3(6): p. 2454-2484.
12. Goriparti, S., et al., Review on recent progress of nanostructured anode materials
for Li-ion batteries. Journal of Power Sources, 2014. 257(0): p. 421-443.
13. Chikkannanavar, S.B., D.M. Bernardi, and L. Liu, A review of blended cathode
materials for use in Li-ion batteries. Journal of Power Sources, 2014. 248(0): p.
91-100.
7. Computer simulations of different battery geometries
165
14. Costa, C.M., M.M. Silva, and S. Lanceros-Mendez, Battery separators based on
vinylidene fluoride (VDF) polymers and copolymers for lithium ion battery
applications. RSC Advances, 2013. 3(29): p. 11404-11417.
15. Roberts, M., P. Johns, and J. Owen, Micro-scaled Three-Dimensional
Architectures for Battery Applications, in Nanotechnology for Lithium-Ion
Batteries, Y. Abu-Lebdeh and I. Davidson, Editors. 2013, Springer US. p. 245-
275.
16. Pikul, J.H., et al., High-power lithium ion microbatteries from interdigitated
three-dimensional bicontinuous nanoporous electrodes. Nat Commun, 2013. 4:
p. 1732.
17. Arthur, T.S., et al., Three-dimensional electrodes and battery architectures. MRS
Bulletin, 2011. 36(07): p. 523-531.
18. Long, J.W., et al., Three-Dimensional Battery Architectures. Chemical Reviews,
2004. 104(10): p. 4463-4492.
19. Pikul, J.H., P.V. Braun, and W.P. King, High power primary lithium ion
microbatteries. Journal of Physics: Conference Series, 2013. 476(1): p. 012087.
20. Sun, K., et al., 3D Printing of Interdigitated Li-Ion Microbattery Architectures.
Advanced Materials, 2013. 25(33): p. 4539-4543.
21. Gaikwad, A.M., A.C. Arias, and D.A. Steingart, Recent Progress on Printed
Flexible Batteries: Mechanical Challenges, Printing Technologies, and Future
Prospects. Energy Technology, 2015. 3(4): p. 305-328.
22. Delannoy, P.E., et al., Ink-jet printed porous composite LiFePO4 electrode from
aqueous suspension for microbatteries. Journal of Power Sources, 2015. 287: p.
261-268.
23. Sousa, R.E., C.M. Costa, and S. Lanceros-Mendez, Advances and Future
Challenges in Printed Batteries. ChemSusChem, 2015. Accepted.
24. Ramadesigan, V., et al., Modeling and Simulation of Lithium-Ion Batteries from
a Systems Engineering Perspective. Journal of The Electrochemical Society,
2012. 159(3): p. R31-R45.
25. Miranda, D., C.M. Costa, and S. Lanceros-Mendez, Lithium ion rechargeable
batteries: State of the art and future needs of microscopic theoretical models and
simulations. Journal of Electroanalytical Chemistry, 2015. 739(0): p. 97-110.
7. Computer simulations of different battery geometries
166
26. Martínez-Rosas, E., R. Vasquez-Medrano, and A. Flores-Tlacuahuac, Modeling
and simulation of lithium-ion batteries. Computers & Chemical Engineering,
2011. 35(9): p. 1937-1948.
27. Franco, A.A., Multiscale modelling and numerical simulation of rechargeable
lithium ion batteries: concepts, methods and challenges. RSC Advances, 2013.
3(32): p. 13027-13058.
28. Arora, P., et al., Comparison between computer simulations and experimental
data for high-rate discharges of plastic lithium-ion batteries. Journal of Power
Sources, 2000. 88(2): p. 219-231.
29. Rohan, J.F., et al., Energy Storage: Battery Materials and Architectures at the
Nanoscale, ICT - Energy - Concepts Towards Zero - Power Information and
Communication Technology, InTech, Editor 2014.
30. Oltean, G., et al., A Li-Ion Microbattery with 3D Electrodes of Different
Geometries. ECS Electrochemistry Letters, 2014. 3(6): p. A54-A57.
31. Zhang, X., Thermal analysis of a cylindrical lithium-ion battery. Electrochimica
Acta, 2011. 56(3): p. 1246-1255.
32. Jeon, D.H. and S.M. Baek, Thermal modeling of cylindrical lithium ion battery
during discharge cycle. Energy Conversion and Management, 2011. 52(8–9): p.
2973-2981.
33. Harb, J.N. and R.M. LaFollette, Mathematical Model of the Discharge Behavior
of a Spirally Wound Lead‐Acid Cell. Journal of The Electrochemical Society,
1999. 146(3): p. 809-818.
34. Hatchard, T.D., et al., Thermal Model of Cylindrical and Prismatic Lithium-Ion
Cells. Journal of The Electrochemical Society, 2001. 148(7): p. A755-A761.
35. Miranda, Á.G. and C.W. Hong, Integrated modeling for the cyclic behavior of
high power Li-ion batteries under extended operating conditions. Applied
Energy, 2013. 111: p. 681-689.
36. Ping, P., et al., Thermal behaviour analysis of lithium-ion battery at elevated
temperature using deconvolution method. Applied Energy, 2014. 129: p. 261-
273.
37. Zhao, R., J. Liu, and J. Gu, The effects of electrode thickness on the
electrochemical and thermal characteristics of lithium ion battery. Applied
Energy, 2015. 139: p. 220-229.
7. Computer simulations of different battery geometries
167
38. Elul, S., Y. Cohen, and D. Aurbach, The influence of geometry in 2D simulation
on the charge/discharge processes in Li-ion batteries. Journal of
Electroanalytical Chemistry, 2012. 682: p. 53-65.
39. Bates, A., et al., Modeling and simulation of 2D lithium-ion solid state battery.
International Journal of Energy Research, 2015: p. n/a-n/a.
40. Lim, C., et al., Geometric Characteristics of Three Dimensional Reconstructed
Anode Electrodes of Lithium Ion Batteries. Energies, 2014. 7(4): p. 2558.
41. Taheri, P., et al., Theoretical Analysis of Potential and Current Distributions in
Planar Electrodes of Lithium-ion Batteries. Electrochimica Acta, 2014. 133: p.
197-208.
42. Doyle, M., et al., Comparison of Modeling Predictions with Experimental Data
from Plastic Lithium Ion Cells. Journal of The Electrochemical Society, 1996.
143(6): p. 1890-1903.
43. Miranda, D., et al., Modeling separator membranes physical characteristics for
optimized lithium ion battery performance. Solid State Ionics, 2015. 278: p. 78-
84.
44. Costa, C.M., et al., Poly(vinylidene fluoride)-based, co-polymer separator
electrolyte membranes for lithium-ion battery systems. Journal of Power
Sources, 2014. 245(0): p. 779-786.
45. Dai, Y., L. Cai, and R.E. White, Simulation and analysis of stress in a Li-ion
battery with a blended LiMn2O4 and LiNi0.8Co0.15Al0.05O2 cathode. Journal
of Power Sources, 2014. 247: p. 365-376.
46. Park, J., et al., Numerical Simulation of the Effect of the Dissolution of
LiMn2O4 Particles on Li-Ion Battery Performance. Electrochemical and Solid-
State Letters, 2011. 14(2): p. A14-A18.
47. Zhang, X., W. Shyy, and A. Marie Sastry, Numerical Simulation of
Intercalation-Induced Stress in Li-Ion Battery Electrode Particles. Journal of The
Electrochemical Society, 2007. 154(10): p. A910-A916.
8. Computer simulation of the effect of different thermal conditions
169
8. Computer simulation of the effect of different thermal
conditions in the performance of conventional and
unconventional lithium-ion battery geometries
This chapter describes the effect of the thermal conditions (isothermal, adiabatic,
cold, regular and hot) in the performance of batteries with conventional, interdigitated,
horseshoe, spiral, ring, antenna and gear geometries. The simulations are based on the
Newman/Doyle/Fuller model with the addition of the thermal model.
This chapter is based on the following publication:
“Computer simulation of the effect of different thermal conditions in the performance of
conventional and unconventional lithium-ion battery geometries”, D. Miranda, C. M.
Costa, A. M. Almeida, S. Lanceros-Méndez, submitted.
8. Computer simulation of the effect of different thermal conditions
171
8.1 Introduction
Electrical energy is increasingly being obtained through renewable sources, such as,
solar, wind, waves, bioenergy and geothermal energy, leading to the need of efficient
energy storage systems [1-4].
These energy storage systems are essential for portable electronic devices such as
mobile phone and computers but also for transportation systems, i.e, power hybrid
electric vehicles (HEVs) and pure electric vehicles (EVs) [5].
Lithium-ion batteries are the most used energy storage systems, being the main type of
battery for many applications [6,7].
Lithium-ion batteries are light weight, show high energy density (210Wh kg-1), low
charge loss, no memory effect, prolonged service-life and high number of
charge/discharge cycles [8,9].
The basic constituents of a lithium-ion battery are the anode, the cathode and the
separator and the main issues for improving its performance are specific energy, power,
safety and reliability [10].
Li-ion batteries are extremely sensitive to certain temperature ranges that depend on
the materials of their constituents and typically can operate between -20 ºC up to ~50-60
ºC [11,12]. The cycling performance of the battery increases with increasing
temperature but if the limit range of temperature is exceeded, exothermic reactions can
occur, increase of the internal pressure, and rupture or even explosion of the battery
[13,14].
For certain applications, such as when high discharge rates are needed for short
operation time, the thermal management of batteries is fundamental to optimize battery
performance [2,14].
The influence of the thermal conditions in lithium ion battery performance is
analyzed in each components but also through of heat dissipation systems [15]. The
electrode thickness influences the battery in many key aspects such as its performance
and overall heat generation [16].
Each active material has different ionic and electrical conductivity values and its
size strongly affects the generation of heat [17].
Thus, the effect of particle size for LiMn2O4 was studied by using a thermal model
and the higher generation of heat generation was observed for larger particles size [18].
Further, the geometry of the batter also influences its thermal behavior [14,19].
8. Computer simulation of the effect of different thermal conditions
172
The thermal behavior of batteries with cylindrical, prismatic and pouch cell
geometries was analyzed under different electrical loads and cooling conditions [20].
In relation to cylindrical cell geometries, it is observed an decreasing heat transfer
resistance with increasing radius due to adiabatic condition at the cell core. On the other
hand, differential temperature across the cell thickness must be considered for prismatic
cells [20].
The thermal behavior of a lithium ion battery during galvanostatic discharge was
analyzed by computer simulation showing that higher cell temperatures raise the risk of
thermal runaway and more rapid degradation of the cell [21].
Due to the relevance of maintaining proper battery temperatures, thermal
management system (TMS) are implemented with the objective to avoid overheating of
battery packs [22]. Applied cooling systems include air cooling [23], liquid cooling
[24], heat pipe cooling [25], and PCM cooling [26].
New lithium-ion unconventional battery geometries, such as ring, spiral, horseshoe,
antenna and gear, [19] can be produced by printing techniques for better integration in
small, portable and wearable devices.
Taking into account the relevance of the thermal behavior of lithium-ion batteries,
the goal of this work is to evaluate the effect of different thermal conditions, including
isothermal, adiabatic and environmental conditions, in the performance of batteries with
seven different geometries: conventional, interdigitated, horseshoe, spiral, ring, antenna
and gear [19].
8. Computer simulation of the effect of different thermal conditions
173
8.2 Preparation and measurement of the full-cell
For the validation of the theoretical thermal model, a full-cell was developed.
For the preparation of the electrodes, anode and cathode, carbon coated lithium iron
phosphate, C-LiFePO4 (LFP, Particle size: D10=0.2 μm, D50=0.5 μm and D90=1.9
μm), poly(vinylidene fluoride) (PVDF, Solef 5130) and N,N’-dimethyl propylene urea
(DMPU) were acquired from Phostech Lithium, Solvay and LaborSpirit, respectively.
Timrex SLG3 graphite particles and carbon black (Super P-C45) were obtained from
Timcal Graphite & Carbon.
The electrodes were prepared by mixing LFP (for the cathode) or graphite (for the
anode) as active materials, Super P, and the PVDF polymer binder in DMPU solvent
with a weight ratio of 80:10:10 (wt.%), i.e, 1 g of solid material for 2.25 mL of DMPU
[27].
First, the polymer was dissolved in the solvent and, after this process, small amounts
of dried mixed solid material (LFP or graphite and Super P) were added to the solution
under constant stirring at room temperature. Good dispersion was achieved by
maintaining the electrode slurry under stirring for 2 hours at 1000 rpm, then 1h in an
ultrasonic bath and then stirred again for 1 hour. After the mixing process of the
materials, the electrode slurry was spread onto a copper foil for the anode and aluminum
foil for the cathode and dried in air atmosphere at 80 ºC in a conventional oven (ED 23
Binder). Finally, the electrodes were dried at 90 ºC in vacuum over the night before
being transferred into the glove-box.
Two Swagelok type cells were assembled in a home-made argon-filled glove box:
the graphite based electrodes (8 mm diameter) were used as anode material; glass
microfiber separators (Whatman grade GF/A) (10 mm diameter) were used as
separators; 1M LiPF6 in ethylene carbonate-diethyl carbonate (EC-DEC, 1:1 vol)
(Solvionic) was used as electrolyte and LFP based electrodes were used as cathode
material (8 mm diameter).
The prelithiation of the graphite electrodes was previously achieved by placing them
in direct contact with an electrolyte-wetted Li foil for 2 hours, under slight pressure. The
active mass loading of the anode and cathode used in the full cell were ~ 1.20 and 2.92
mg.cm-2, respectively.
The full battery was cycled at 25 ºC from 2.5 V to 3.8 V at C/10 rate (C = 170
mA.g-1) using a Landt CT2001A Instrument.
8. Computer simulation of the effect of different thermal conditions
174
8.3 Theoretical model: parameters, initial values and boundary conditions
8.3.1 Theoretical simulation model
Simulations were performed by applying the electrochemical model based on the
Newman/Doyle/Fuller equations with addition of the thermal behaviour. This
electrochemical model describes the electrochemical processes that occurs in battery
components, electrodes, separator and current collectors, including the thermal
behaviour. The simulations were carried out by implementing the finite element method
for different 2D battery geometries in a typical lithium-ion battery structure: [porous
negative electrode, (LixC6) | porous membrane of glass micro fiber soaked in 1M
lithium hexafluorophosphate (LiPF6) in ethylene carbonate-diethyl carbonate (EC-DEC)
| porous positive electrode, (LixFePO4)]. The degree of porosity of the electrodes,
defined as the space between the particles of active electrode material, is shown in table
8.1 for both electrodes.
The finite element calculations describing the electrochemical and thermal behavior
were carried out using a MATLAB subroutine to solve the main equations describing
the behavior of the different battery constituents (electrodes and separator) in an ideal
cell without solid electrolyte interface (SEI) formation, as presented in Chapter 3. The
size of the mesh was refined according to the dimensions of the different geometries of
the battery.
The value of C-rate was determined by the area of the cathode considering the mass
of active material.
The impedance was measured for each geometry at frequencies ranging from 10
MHz to 1 MHz with a potential perturbation amplitude of 0.01 V and with the following
parameters: film resistance of the positive electrode: 0.0065 m2·S-1; film resistance of
the negative electrode: 1×10-5 m2·S-1; double layer capacitance of the positive electrode:
0.2 F.m-2; double layer capacitance of the negative electrode: 0.2 F.m-2; current collector
resistance at each current collector: 1.1×10-4 m2.S-1 [28].
8. Computer simulation of the effect of different thermal conditions
175
8.3.2 Specific parameters and initial values
The values of the parameters used for the different components of each battery
geometry are listed in Table 8.1. The areas of all components were maintained constant
in the computer simulations as shown in [29].
Table 8.1 - Values of the parameter values used in the simulations. The nomenclature is
indicated in the List of Symbols and Abbreviations.
Electrochemical parameters and initial values
Parameter Unit Anode (LixC6) Separator Cathode (LixFePO4)
CE,i,0 mol/m3 14870 3900
CE,i,max mol/m3 31507 21190
CL mol/m3 1000
r m 12,510-6 810-6
Li m 20010-6 9010-6 40010-6
ki(T) S/m a) a) a)
kef,i S/m ki(T) 0,3571,5 ki(T) 4,8410-2 ki(T) 0,4441,5
Kt298,15,i m/s 210-11 210-11
Kt,i (T) m/s b) b)
Di(T) m2/s c) c) c)
Def,i m2/s Di(T)0,3571,5 Di(T)4,8410-2 Di(T)0,4441,5
DLI m2/s 3,910-14 3,210-13
DLI(T) d) d)
Brugg or p 1,5 8,5 1,5
f,i 0,172 0,259
i 0,357 0,70 0,444
3,8
i S/m 100 11.8
i1C
A/m2 17,5
F C/mol 96487
R J/mol K 8,314
Ead,i J/mol 5.1103 39103
Eak,i J/mol 58103 29103
Thermal parameters and initial values
Parameter Unit Anode (LixC6) Separator Cathode (LixFePO4)
Cp,i J/(kg.K) 1437.4 1978.16 1260.2
i kg/m3 2660 1008.98 1500
i W/(m.K) 1,04 0,344 1,48
h W/(m2.K) 1,0 1,0 1,0
T,cold K 265,15 265,15 265,15
T, reg K 298,15 298,15 298,15
T,hot K 316,15 316,15 316,15
T0,adi K 298,15 298,15 298,15
T0,cold K 265,15 265,15 265,15
T0,reg K 298,15 298,15 298,15
T0,hot K 316,15 316,15 316,15
Area of each component of the battery
Parameter Unit Anode (LixC6) Separator Cathode (LixFePO4)
Ai m2 4,010-8 1,810-9 8,010-8
8. Computer simulation of the effect of different thermal conditions
176
Auxiliary equations:
a) Ionic conductivity as a function of temperature [30]:
ki(T) = c (-10.5+(0.0740T)-((6.9610-5) (T2))+(0.668c)-
-(0.0178cT)+((2.810-5)c (T2))+(0.494c2)-((8.8610-4) (c2)*(T)))2
b) Reaction rate coefficient of the electrodes as a function of temperature [31]:
Kt,i (T)= kt298,15,i exp(-(Eak,i/R) (1/T-1/298,15))
c) Diffusion coefficient of the salt in the electrolyte as a function of temperature
[30]:
Di(T) = 10^(-(0.22c)-4.43-((54)/(T-229-(5c))))
d) Diffusion coefficient of Li ions in the electrode as a function of temperature [31]:
DLI(T) = DLI exp(-(Ead,i/R) (1/T-1/298,15))
Table 8.2 shows the schematic representation of each of the evaluated geometries,
conventional, interdigitated, gear, horseshoe, spiral, antenna and ring, as well as the
values of the relevant dimensions for battery characterizations such as distance between
collectors and thickness of the electrodes and separator, among others.
8. Computer simulation of the effect of different thermal conditions
177
Table 8.2 - Schematic representation of the different battery geometries and the
corresponding dimensions. The nomenclature is indicated in the List of Symbols and
Abbreviations.
Battery design
Dimensions (m)
Parameter
Value / m
Conventional
Lc 400×10-6
La 200×10-6
e_sep 90×10-6
d_max 697×10-6
d_cc 690×10-6
Interdigitated
Parameter Value / m
N 8 digits
c_dig 100×10-6
e_dig 20×10-6
e_sep 8.66×10-6
d_max 391×10-6
d_cc 327×10-6
8. Computer simulation of the effect of different thermal conditions
178
Gear
Parameter Value / m
N 8 digits
e_sep 12.41×10-6
Rg 93.9×10-6
d_max 294×10-6
d_cc 135.8×10-6
e_dig 40×10-6
c_dig 30×10-6
Horseshoe
Parameter Value / m
Lc 33,1×10-6
La 17.5×10-6
e_sep 7.71×10-6
d_max 1125×10-6
d_cc 58.3×10-6
Spiral
Parameter Value / m
Lc 28.6×10-6
La 17.8×10-6
e_sep 7.27×10-6
d_max 1240×10-6
d_cc 53.7×10-6
8. Computer simulation of the effect of different thermal conditions
179
Antenna
Parameter Value / m
Lc 25.6×10-6
La 16.0×10-6
e_sep 5.88×10-6
d_max 1225×10-6
d_cc 47.5×10-6
Ring
Parameter Value / m
Lc 27.4×10-6
La 14.5×10-6
e_sep 6.40×10-6
Rd 430×10-6
d_max 1350×10-6
d_cc 48.4×10-6
8. Computer simulation of the effect of different thermal conditions
180
8.3.3 Boundary conditions
The boundary conditions were defined accordingly to the electrochemical
(Newman/Doyle/Fuller) and thermal models. The boundary conditions are
schematically presented in Figure 1 and defined in Table 8.3.
As the boundary conditions are the same for all geometries, just the ones for
conventional geometry will be presented, as an example. Table 4 shows these boundary
conditions addressing the schematic representation of the conventional geometry,
illustrated in Figure 8.1.
In Table 8.3 and Figure 8.1, the boundary conditions are identified from 1 to 7.
According to Figure 8.1, the boundary condition 1 indicates that there is no ion
flux. Regarding the thermal model in the adiabatic condition, there is no heat transfer
with the external environment, as defined by boundary condition 1. Also, at adiabatic
condition the external temperature is not applicable (boundary condition 6). In contrast,
for the thermal model with different conditions (cold, regular and hot temperatures)
there is heat transfer with the external environment (boundary condition 1) and an
external temperature was defined according to the applied thermal condition (boundary
condition 6).
For the interfaces between the electrodes and the separator, as well as between the
electrodes and the current collectors/external medium, the boundary conditions 2, 3, 4, 5
were defined. These boundary conditions define the value of the ionic
conductivity/diffusion, concentration of lithium ions and electric conductivity for both
sides of the interface border.
Finally, the boundary condition 7 defines the values of the ionic diffusion and
concentration of the lithium salt along the radius of the spherical particles of active
material.
8. Computer simulation of the effect of different thermal conditions
181
Table 8.3 - Summary of the boundary conditions implemented in the conventional
geometry. The nomenclature is indicated in the List of Symbols and Abbreviations.
Boundaries Boundary Condition Model
Boundary 1 No ion flux occurs.
Electrochemical
model
(Newman, Doyle,
Fuller)
Boundary 2
02
,
,
x
aL
aefx
cD
02
,
,
x
aL
aefx
k
Boundary 3
3
,
,
3
,
,
x
sL
sef
x
aL
aefx
cD
x
cD
3,3, xsLxaL cc
03
,
,
x
aE
aefx
Boundary 4
4
,
,
4
,
,
x
cL
cef
x
sL
sefx
cD
x
cD
4,4, xcLxsL cc
04
,
,
x
cE
aefx
Boundary 5
05
,
,
x
cL
cefx
cD
app
x
cE
cef ix
5
,
,
05
,
,
x
cL
cefx
k
Boundary 7
cai
Jr
cDrrAt
r
cDrAt
i
iE
iLii
iE
iLi
,
:
0:0
,
,
,
,
Boundary 1 csaiTxi ,,,0
1
Thermal Model
(Adiabatic condition) Boundary 6 T , the external temperature is not applicable.
8. Computer simulation of the effect of different thermal conditions
182
Boundary 1 TThTxi 1
Thermal Model
(cold, regular and hot
temperatures)
Boundary 6 T , the external temperature according to the
thermal conditions applied.
Figure 8.1 - Schematic representation of the boundary conditions applied in the
conventional geometry.
8. Computer simulation of the effect of different thermal conditions
183
8.4 Results and discussion
Theoretical model simulations were thus applied in all different lithium-ion battery
geometries in different thermal conditions: isothermal, adiabatic and environmental
conditions (cold, regular and hot temperatures) keeping constant the area of the
components.
The theoretical model was first validated with the experimental results obtained for
the developed full cell.
The main objective is to evaluate how the performance of the batteries with different
geometries are affected by the thermal conditions.
8.4.1 LiC6/LiFePO4 full-cell: Validation of the theoretical model
The simulation model was validated by comparing the theoretical and experimental
results obtained for the LiC6/LiFePO4 full-cell with conventional geometry (figure 8.2).
Figure 8.2 shows experimental and simulation curves for the full-cell at 298 K and at
scan rate of C/10 (0.51 A.m-2).
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
2,4
2,6
2,8
3,0
3,2
3,4
3,6
Experimental result
Theoretical result
Voltage
/ V
Capacity / Ah.m-2
Figure 8.2 - Voltage as a function of the delivered capacity at C/10 rate for the
LiC6/LiFePO4 full-cell with a conventional geometry.
Figure 8.2 shows a good agreement between experimental and theoretical results.
There is a slight deviations of the capacity of the real full-cell relative to the theoretical
8. Computer simulation of the effect of different thermal conditions
184
model below 3.2 V, attributed to corresponding differences in the electronic
conductivity values and also to the exact temperature value during the discharge process
[31].
However, the good theoretical approximation allows the validation of the theoretical
model.
8.4.2 Battery performance of the various battery geometries at different thermal
conditions
Theoretical model simulations with thermal conditions were carried out for all
geometries at different thermal conditions (isothermal, adiabatic and environmental
(cold, regular and hot temperatures) varying scan rate between 1C to 500C.
8.4.2.1 Isothermal condition
Firstly, all geometries were tested at scan rates between lC to 500C for a constant
temperature of 298 K, i.e, without applying the thermal equations.
A similar study has been already presented for all these geometries with lithium
manganese oxide (LiMn2O4, LMO) as active material [19], which leads to differences
based on the specific electric and ionic conductivity values and lithium diffusion
coefficients, among others, of the active materials [33].
Thus, the present investigations also allow to evaluate the influence of active
material on battery performance.
Figure 8.3 shows the discharge capacity value as a function of the scan rate for all
geometries under isothermal condition.
8. Computer simulation of the effect of different thermal conditions
185
0 100 200 300 400 500
0
100
200
300
400
500
600
700
800
Ca
pa
city / A
h.m
-2
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.3 - Delivered capacity as a function of scan rate for all geometries under
isothermal condition.
For low scan rates, no significant differences arise and all geometries show high
capacity. At medium and high scan rates, the discharge capacity of the geometries
follow this order: interdigitated, gear, horseshoe, spiral, antenna, ring and conventional.
It can be observed that the conventional geometry operates just up to 300C and for
this scan rate, its discharge capacity value is 3.61 Ah.m-2. It is also observed that all
other geometries may operate at higher rates up to 500C.
The interdigitated geometry shows the best performance for all scan rates. At 300C,
its capacity is 356 Ah.m-2, which is 98 times higher than the one for the conventional
geometry.
The gear geometry closely follows the interdigitated one and at 300C, the capacity
value is 354 Ah.m-2.
The different discharge capacity values observed for the geometries is ascribed to
different internal resistance values, variations of the maximum distance, and distance
between current collectors in the different geometries, as well as to variations of the
dimensions of the components (electrodes and separator), as shown in table 8.2.
Thus, the main reason for the conventional geometry not operating at scan rates
above 300C is due to the high thickness of the electrodes and separator in comparison to
the other geometries, which limits the diffusion of ions.
8. Computer simulation of the effect of different thermal conditions
186
The interdigitated and gear geometries show lower maximum distance values
(d_max) and higher contact surface area between the electrodes than any other
geometry, as can be seen in table 8.2.
During the discharge process, the geometry effect is more significant for higher scan
rates once it is required elevated mobility of ions and electrons.
By comparison with the literature [19], it is observed that the results with LiFePO4
or LiMn2O4 as active material are similar.
8.4.2.2 Adiabatic condition
All geometries were tested under adiabatic condition with an initial temperature of
298.15 K before the discharge process for all geometries. Figure 8.4a) and 8.4b) show
the discharge capacity value and the temperature for all geometries at scan rate between
1C and 500C, respectively.
0 100 200 300 400 500100
200
300
400
500
600
700
800a)
Ca
pa
city / A
h.m
-2
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 100 200 300 400 500295
300
305
310
315
320
b)
Tem
pe
ratu
re / K
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.4 - Delivered capacity (a) and temperature (b) as a function of the scan rate for
all geometries under adiabatic condition.
The discharge capacity value decreases when increasing the scan rate as it is shown
in figure 8.4a). Up to 200 C, the discharge capacity value is practically the same for all
geometries, being the differences observed in the discharge capacity value after 300 C
for the different geometries attributed to variation in the internal resistance of the
batteries due to geometrical effects, as previously indicated.
The discharge capacity value is higher under adiabatic conditions when compared
to isothermal condition for the same scan rates. The reason for this effect is due that
8. Computer simulation of the effect of different thermal conditions
187
heat produced for each geometry will be internally absorbed, leading to a temperature
increase (figure 8.4b)), which in turn affects the diffusion and ionic conductivity values
(Chapter 3, Table 3.1).
As previously observed, the conventional geometry only operates up to 300C and
its discharge capacity value is 367.05 Ah/m2. For this geometry, the discharge value in
the adiabatic condition is higher relative to the isothermal condition (3.61 Ah/m2), due
to the increase of temperature and the corresponding effect on the diffusion and ionic
conductivity values [34].
The conventional geometry has higher internal resistance due to longer distance
between current collectors and larger thickness of the separator. Thus, it is observed that
the higher discharge capacity caused by the increase of the temperature it is not
sufficient to overcome the losses associated to the high internal resistance.
Figure 8.4b) shows that the battery temperature increases with the scan rate up to
300 C for all geometries, due to the heat produced by ohmic losses [35]. For scan rates
above 300 C it is observed that the temperature decreases as the scan rate increase, as
the heat produced is not totally absorbed during the discharge cycle due to the low
discharge time.
Figure 8.4b) also shows that the interdigitated and gear geometries present lower
temperature values relatively to the other geometries. The main reason for this behavior
is due to the smaller separator thickness, lower maximum distances that ions cross until
their intercalation (d_max) and higher contact surface between the electrodes.
In adiabatic conditions, the interdigitated geometry shows higher discharge capacity
value for all scan rates in comparison to the other geometries, including the
conventional geometry. As for the temperature value, the conventional geometry has
higher value when compared to other geometries due to its higher internal resistance.
To evaluate the internal resistance of the battery, impedance measurements were
carried out for the conventional and interdigitated geometries at 298.15 K.
Figure 8.5 shows the Nyquist plot for both geometries in the adiabatic condition at
frequencies between 1 MHz to 0.1 mHz.
8. Computer simulation of the effect of different thermal conditions
188
0 5x10-5
1x10-4
2x10-4
2x10-4
0
2x10-5
4x10-5
6x10-5
8x10-5
1x10-4
1x10-4
1x10-4
-Z''
/
.m2
Z' / .m2
Conventional
Interdigitated
Figure 8.5 - Nyquist plot for conventional and interdigitated geometries under adiabatic
condition.
The Nyquist plots are composed of semicircles (overall resistance) at higher and
medium frequency and a straight line at lower frequencies [36] as it is illustrated in
figure 8.5. Figure 8.5 shows that the conventional geometry shows higher internal
overall resistance when compared to the interdigitated geometry. The internal overall
resistance is 1.10 × 10-4 Ω.m2 and 5.94 × 10-5 Ω.m2 for conventional and interdigitated
geometries, respectively.
Thus, considering the thermal model, the battery performance is a balance between
the higher discharge capacity value caused by the increase of the temperature and the
losses related to the internal overall resistance.
8.4.2.3 Environmental conditions
All geometries were subjected to three thermal external conditions considering
initial thermal equilibrium with the environmental, whose temperature is cold, 265.15
K, figures 8.6a) and 8.6b); regular, 298.15 K, figures 8.6c) and 8.6d); and hot, 316.15
K, figures 8.6e) and 8.6f).
For each case, the heat produced during the discharge process is exchanged with the
exterior and for each geometry, the discharge capacity, the total heat (irreversible,
reversible and ohmic heat) and internal temperature were evaluated as a function of the
scan rate.
8. Computer simulation of the effect of different thermal conditions
189
Figures 8.6a) and 8.6b) show the discharge capacity value and temperature as a
function of the scan rate between C at 250 C, respectively, for cold condition and all
geometries.
None of the batteries can operate at scan rates above 250 C, as the low temperature
(265.15 K) severely limits the diffusion and the ionic conductivity of the electrolyte
solution [17]. The battery performance for all geometries is identical to the one
observed for isothermal and adiabatic condition but with lower discharge capacity
values. The conventional geometry only operates up to a scan rate of 17 C, for which
the discharge capacity value is 454 Ah.m-2.
At 250 C the discharge capacity values are 234 Ah.m-2, 194 Ah.m-2, 158 Ah.m-2,
212 Ah.m-2, 280 Ah.m-2 and 319 Ah.m-2 for horseshoe, spiral, antenna, ring,
interdigitated and gear geometries, respectively. The interdigitated geometry shows the
best performance under isothermal and adiabatic condition but for cold conditions the
values for both geometries are very close. However, the gear geometry shows slightly
better performance than the interdigitated geometry as the gear geometry has smaller
distance between current collectors and lower distances for ions to move until their
intercalation.
0 50 100 150 200 250100
200
300
400
500
600
700T
i=263.15K
a)
Ca
pa
city / A
h.m
-2
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 50 100 150 200 250264
266
268
270
272
274
276
Ti=263.15Kb)
Te
mp
era
ture
/ K
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
8. Computer simulation of the effect of different thermal conditions
190
0 50 100 150 200 250 300 350 400 450 500 550
100
200
300
400
500
600
700T
i=298.15K
c)
Ca
pa
city / A
h.m
-2
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 50 100 150 200 250 300 350 400 450 500 550
300
305
310
Ti=298.15Kd)
Tem
pe
ratu
re / K
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 50 100 150 200 250 300 350 400 450 500 550
200
300
400
500
600
700 Ti=316.15K e)
Ca
pa
city / A
h.m
-2
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 50 100 150 200 250 300 350 400 450 500 550
316
318
320
322
324
326
328
Ti=316.15K
f)
Tem
pe
ratu
re / K
Scan rate / C
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.6 - Delivered capacity (left) and final temperature (right) as a function of the
scan rate for all geometries under cold (a and b), regular (c and d) and hot (e and f)
conditions.
In relation to the temperature (figure 8.6b)), the conventional geometry reaches
higher temperatures in comparison to the other geometries due to its higher internal
overall resistance. The higher temperature value observed for interdigitated and gear
geometries is due to the lower exchange of heat with the exterior when compared to the
other geometries resulting in higher discharge capacity values. The contact area between
the battery and the exterior for the interdigitated and gear geometries is lower, affecting
therefore the heat transfer process.
Figures 8.6c) and 8.6d) show the discharge capacity and temperature values,
respectively for all geometries as a function of the scan rate (C to 500 C) for regular
environmental condition (298.15K).
8. Computer simulation of the effect of different thermal conditions
191
For this thermal condition, the conventional geometry can operate up to 300 C and
all other geometries up to 500 C. At 300 C, the conventional geometry has the lowest
discharge capacity value (318 Ah.m-2) and interdigitated geometry has the highest
discharge capacity value (371 Ah.m-2) in comparison to the other geometries.
As it was observed under adiabatic condition, the increases of the diffusion and
conductivity value due to the increase of temperature value leads to an increase in
battery discharge capacity. It is to notice that the horseshoe, spiral, antenna and ring
geometry does not reflect the increase in battery performance once these geometries
have elevated distances for ions and electrons to move during the discharge process.
Figure 8.6c) shows that the conventional geometry presents highest temperature
value once it absorbs the heat produced due its internal overall resistance value. For this
thermal condition, it is also observed that interdigitated and gear geometries have higher
temperature value in comparison to the horseshoe, spiral, antenna and ring geometries
for the same reason that was observed for the cold condition.
Figures 8.6e) and 8.6f) show the discharge capacity and temperature values,
respectively, for all geometries as a function of the scan rate (1C to 500 C) for hot
condition (316.15 K).
For this thermal condition, differences in discharge capacity value are just observed
at scan rates above 400 C for all geometries. For this temperature, the conventional
geometry operates up to 500 C. Further, the differences in the discharge capacity values
are small in comparison to the isothermal, adiabatic, cold and environmental conditions
for all geometries
As previously observed, the interdigitated and gear geometries show the best
discharge values in comparison to the other geometries. Relatively to the temperature
behavior (figure 8.6f)), the conventional geometry has the higher temperature value for
all scan rates due to the higher separator thickness. The temperature behavior for the
other geometries (figure 8.6f)) is the same as observed for adiabatic and environmental
conditions.
It is important refer that the conventional geometry reach temperature above 323 K
for scan rates above 300 C where the organic solvent of the electrolyte solution can start
to evaporate [37].
8. Computer simulation of the effect of different thermal conditions
192
8.4.3 Total heat at low and high discharge rates
The total dissipated heat for the different geometries was evaluated with the
objective to relate the increases of the temperature with the total heat produced by the
battery.
The generated heat in the battery comes from three sources: reaction, reversible and
ohmic. The total heat of the different components (anode, separator and cathode) was
determined for all geometries under adiabatic conditions at low scan rate (1C) and high
scan rate (300C) once the conventional geometry only operates up to this scan rate.
Figure 8.7a), 8.7b) and 8.7c) show the total heat in the anode, separator and
cathode, respectively, for all geometries at 1C as a function of time.
0 30000 60000 90000 120000 150000-300
-200
-100
0
100
200
300
400 a)
QT
ota
l / W
.m-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 30000 60000 90000 120000 1500000
2
4
6
8
10
b)
0 30000 60000 90000 120000 150000
-0,2
0,0
0,2
0,4
0,6
0,8
1,0
QT
ota
l d
issip
atio
n / W
.m3
Time / s
Q
To
tal /
W.m
-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 30000 60000 90000 120000 150000
0
20
40
60
80
100
120
c)
QT
ota
l / W
.m-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0,0000 0,0002 0,0004 0,0006 0,0008
0
20
40
60
80
100 d)
QT
ota
l / W
.m-3
x / m
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.7 - Total heat in the anode (a), separator (b) and cathode (c) for all geometries
at 1C as a function of the time. d) Total heat along the battery for all geometries at 1C
after 120 000s.
8. Computer simulation of the effect of different thermal conditions
193
The total discharge time for all geometries at 1C is around 150 000s. Figure 8.7d)
shows the total heat for all geometries at different places between current collectors at
time of 120 000s
Figure 8.7a) and 8.7c) show that the total heat produced by the electrodes (anode
and cathode, respectively) is the same at all instants of time along the discharge cycle.
For all geometries, the total dissipated heat for anode changes between -220 W/m3 to
370 W/m3 and for cathode varies between 0 W/m3 to 110 W/m3, as a function of time.
All geometries produce the same amount of heat in each electrode, being therefore
identical the effect of losses associated with the internal resistance caused by the
diffusion and conductivity of the ions and the electrical conduction.
The heat produced is the same in all geometries as at low discharge rates, a low ionic
mobility is required and the internal resistance has not significant effects in the
produced heat.
For the separator, figure 8.7b) shows that the total dissipated heat for conventional
geometry is higher in comparison to the other geometries. Thus, for the conventional
geometry the varies from 8.8 W/m3 to 9.5 W/m3 as a function of time and for other
geometries varies between 0.01 W/m3 to 0.56 W/m3.
This fact is due to the higher separator thickness for the conventional geometry in
comparison to the other geometries that affects the mobility of the ions and in turn the
produced heat.
Figure 8.7d) shows the total heat at a time of 120 000 s in different points between
current collectors for all geometries, where the heat is produced according the results of
the figure 8.7a) to 8.7c).
As represented in figure 8.7d) at 1C, the total heat of the electrodes is very close for
all geometries, the difference being verified for the separator due to their thicknesses.
Further, the evolution of the temperature of the battery as a function of time for all
geometries is shown in figure 8.8 at 1C.
8. Computer simulation of the effect of different thermal conditions
194
0 30000 60000 90000 120000 150000296
297
298
299
Te
mp
era
ture
/ K
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.8 - Temperature of the battery as a function of time for all geometries at 1C.
Figure 8.8 shows that the temperature of the batteries along to the discharge time is
independent of the geometry.
Figures 8.9a) 8.9b) and 8.9c) show the total dissipated heat as a function of time in
the anode, separator and cathode, respectively, for all geometries at 300 C. This
condition is selected at the higher scan rate will be produce a larger effect of the internal
resistance during to the discharge process.
Figure 8.9a) shows that the geometries that produce a lower amount of heat in the
anode are the gear and the interdigitated geometries.
In contrast, the conventional, ring, spiral and antenna geometries produce larger
amounts of heat along to the discharge time.
8. Computer simulation of the effect of different thermal conditions
195
0 100 200 300
1,0x105
1,5x105
2,0x105
2,5x105
3,0x105 a)
QT
ota
l / W
.m-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 80 160 2400
1x104
2x104
3x104
4x104
4x105
6x105
8x105 b)
QT
ota
l / W
.m-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
0 80 160 240
5,0x104
1,0x105
1,5x105
2,0x105
2,5x105
3,0x105
c)
QT
ota
l / W
.m-3
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.9 - Total heat for anode (a), separator (b) and cathode (c) for all geometries at
300C as a function of time.
It is to notice that, the antenna, ring and spiral geometries show small distance
between current collectors and also small separator thickness as well as longer distances
for the ions to move (d_max) that implies higher dissipated heat due to ohmic losses. It
is interesting to notice that close to the end of the discharge time, the ring, antenna,
spiral and horseshoe geometries approach to the conventional geometry behavior since
the total dissipated heat for these geometries increases over time due to the contribution
of the ions located in places more distant from the collectors.
Identical behavior is observed for the cathode (figure 8.9c)), where the
conventional geometry shows higher total dissipated heat (270 kW/m3) in comparison to
the other geometries and the geometry with the lower total dissipated heat is the gear
geometry (50 kW/m3).
8. Computer simulation of the effect of different thermal conditions
196
Relatively to the separator (figure 8.9b)), the conventional geometry also shows
higher total dissipated heat in comparison to the other geometries, with values between
388 kW/m3 and 810 kW/m3. The interdigitated and gear geometries show intermediate
values of 39 kW/m3 and 14 kW/m3, whereas the dissipate heat is between 1071W/m3 to
7200W/m3 for the other geometries. In this case, the conventional geometry shows
higher total dissipated heat due to the higher separator thickness in comparison to the
other geometries (Table 8.2) and the interdigitated and gear geometries show
intermediate values due to the separator thickness and distance between current
collectors.
Figures 8.10a) and 8.10b) show the total heat in different positions on the battery
between the current collectors for all geometries after 50 s at 300C.
0,0000 0,0002 0,0004 0,0006 0,0008
0
9x104
2x105
3x105
4x105
5x105
5x105
6x105
7x105
a)
QT
ota
l / W
.m-3
x / m
conventional
interdigitated
0,00000 0,00004 0,00008 0,00012
0
2x104
4x104
6x104
8x104
1x105
1x105
1x105 b)
QT
ota
l / W
.m-3
x / m
horseshoe
spiral
ring
antenna
gear
Figure 8.10 - Total heat along the battery after 50 s at 300C for conventional and
interdigitated geometries (a) and for the remaining geometries (b).
Figure 8.10a) shows that batteries with conventional geometry generate higher heat
due to the thickness of the separator. Figure 8.10 also shows that the interdigitated and
gear geometries generate lower heat values in all positions between the current
collectors due to the lower thickness of separator.
The geometries that produced lower total dissipated heat are the interdigitated and
gear geometries associated to internal resistance related to the thickness of the separator,
distance between current collectors and smaller distance of ions until intercalation.
Figure 8.11 shows the evolution of the temperature of the battery as a function of
time with the different geometries at 300C.
8. Computer simulation of the effect of different thermal conditions
197
0 50 100 150 200 250 300
300
310
320
Te
mp
era
ture
/ K
Time / s
conventional
interdigitated
horseshoe
spiral
ring
antenna
gear
Figure 8.11 - Temperature as a function of time for all geometries at 300C.
Figure 8.11 shows that the temperature of the different batteries increases linearly in
time. The conventional geometry temperature increases from 298.15 K to 318.10 K,
whereas for the interdigitated and gear geometries the increase is the lowest reaching a
maximum of 310 K after 260 s discharge time.
8.4.4 Ohmic heat for ring geometry with different radius
For understanding the contribution of the mobility of the ions and the
corresponding resistance for the production of heat and its influence into battery
performance, this section analyzes the ohmic heat produced in a ring geometry for
different radius at 500C and under adiabatic conditions. The selected radius are 93.9
m, 230 m, 330 m and 430 m (Figure 8.12)), the ohmic heat being associated to the
Joule effect caused by the ohmic losses due to the different paths the charges have to
move to the current collectors (d_max) and also to differences in the thickness of the
separator. The variation of the radius is carried out while maintaining constant the area
of all components of battery (electrodes, separator and current collectors).
8. Computer simulation of the effect of different thermal conditions
198
Figure 8.12 - Schematic representation of the ring geometry for the radius of 93.9 m
and 430 m.
Simulations show (Figure 8.13 a)) that the capacity of the battery decreases with
increasing the radius of the ring. The ring geometry with bigger radius has smaller
thickness for electrodes and separator, lower distance between the current collector, and
higher maximum distance for the ions (d_max) that implies lower battery performance
due to the higher internal resistance.
It is observed in figure 8.13b) that the temperature of the battery increases over
time for all radii, being the increase larger for the larger radius.
50 100 150 200 250 300 350 400 450180
190
200
210
220
230
240
250
260
270
Ca
pa
city / A
h.m
-2
Ring radius / m
a)
0 20 40 60 80 100 120
298
300
302
304
306
308
310 b)
Te
mp
era
ture
/ K
Time / s
r = 93.9m
r = 230m
r = 330m
r = 430m
Figure 8.13 - a) Capacity as a function of ring radius and b) temperature as a function
of time for all ring radius at 500 C.
8. Computer simulation of the effect of different thermal conditions
199
The heat for each component (electrodes and separator) as a function of time for the
batteries with the different radius is shown in figure 8.14.
Figure 8.14 a) and 8.14c) show that for both electrodes (anode and cathode) the
generated heat increase with increasing radius.
As previously indicated, higher ohmic losses are observed for the geometry with
the larger radius due to the increase of the path the charges have to move during the
discharge process.
0 20 40 60 80 100 1200
2x104
4x104
6x104
8x104
1x105
1x105
a)
Qo
hm
ic / W
.m-3
Time / s
r = 93.9m
r = 230m
r = 330m
r = 430m
0 20 40 60 80 100 1200
2x103
4x103
6x103
8x103
1x104
1x104 b)
Qo
hm
ic / W
.m-3
Time / s
r = 93.9m
r = 230m
r = 330m
r = 430m
0 20 40 60 80 100 120
3x104
4x104
5x104
6x104
7x104
8x104
9x104
c)
Qo
hm
ic / W
.m-3
Time / s
r = 93.9m
r = 230m
r = 330m
r = 430m
Figure 8.14 - Ohmic heat for anode (a), separator (b) and cathode (c) as a function of
the time at 500 C for various ring radius.
Figure 8.14b) shows the ohmic heat at the separator for the ring geometries with
different radius. Contrary to the observations for the electrodes (figure 8.14a) and
8.14c)), the higher ohmic heat is observed for the battery with the smaller radius. In this
case, the ring geometry with smaller radius produce higher ohmic heat due to the higher
separator thickness, that will affect the diffusion and conduction behavior of the ions
and consequently to increase by the Joule effect that translate in higher ohmic heat.
8. Computer simulation of the effect of different thermal conditions
200
Figure 8.15 shows the ohmic heat fin the battery along different positions between
the current collectors for the ring geometry with different radius at 70 s and 500 C.
0,00000 0,00005 0,00010 0,00015
0
2x104
4x104
6x104
8x104
1x105
1x105
Qo
hm
ic / W
.m-3
x / m
r = 93.9m
r = 230m
r = 330m
r = 430m
Figure 8.15 - Ohmic heat along different places between the current collectors of the
battery after 70 s at 500C for ring geometry with different radius.
Considering figure 8.15, it is observed higher ohmic heat for the electrodes of the
ring geometry with higher radius due to the larger distance the ions have to move until
intercalation.
In relation to the separator, the higher ohmic heat is observed for the smaller radius due
to higher separator thickness.
Figure 8.16 shows the impedance curves for the ring geometry with different radius
in order to determine the internal resistance value.
Independently of the radius of the ring geometry, the Nyquist plot is characterized
by two semicircles at high frequencies identified in the figure 8.16 where the overall
resistance that is the sum of the two semicircles that represent the ohmic resistance,
which is related to the contact film resistance and resistance contributions from the
charge-transfer reaction resistance in the high and medium frequency range. At low-
frequency range, the inclined line corresponds to the Warburg impedance, associated to
the lithium-ion diffusion in the bulk of the active material [38]. The diameter of the
semicircles corresponds to the total impedance and its value is 5 × 10-5 Ω.m2, 6 × 10-5
Ω.m2, 8 × 10-5 Ω.m2 and 9 × 10-5 Ω.m2 for 93.9 m, 230m, 330m and 430m,
respectively.
8. Computer simulation of the effect of different thermal conditions
201
0 2x10-5
4x10-5
6x10-5
8x10-5
1x10-4
1x10-4
0
3x10-5
6x10-5
9x10-5
1x10-4
-Z''
/
.m2
Z' / .m2
r = 93.9m
r = 230m
r = 330m
r = 430m
Figure 8.16 - Nyquist plot for the ring geometry with different radius at 500 C.
It is observed that the ring geometry with small radius shows the lowest resistance
value due to the smaller paths that the ions placed distant from current collectors have to
move.
To understand the effect of the maximum distance of ions until the intercalation
process, figure 8.17 shows the ionic current density vectors for the ring geometry with
small (figure 8.17a) and higher (figure 8.17b)) radius that correspond to the ionic charge
at the time of 70 s at 500C. This magnitude is represented as a vector on the 2D
graphics shown in figure 8.17, indicating the direction of the ions.
8. Computer simulation of the effect of different thermal conditions
202
Figure 8.17 - Ionic current density vectors of the ring geometry for a) R= 93.9 m and
b) R=430m.
Figure 8.17a) shows that for the battery with the smaller radius, the most distant
ions have a shorter maximum distance to travel to the current collectors. In relation of
the battery with bigger radius (figure 8.17b)), it is observed a similar behavior. Figure
8.17b) also shows that for places closer to the current collectors, the ionic current
density is more intense, the ring geometry with higher radius dissipating more ohmic
heat.
It is concluded that battery performance for each geometry can be optimized
considering the geometrical parameters that will be influence the thermal behavior.
8. Computer simulation of the effect of different thermal conditions
203
8.5 Conclusions
Thermal conditions are a critical issue in lithium-ion batteries as they influence the
battery performance and safety. For maximizing the battery performance, it is essential
to carry out the geometry optimization considering the thermal modelling. This work
shows the effect of the thermal conditions for different geometries: conventional,
interdigitated, horseshoe, spiral, ring, antenna and gear geometries. The simulations
were based on the Newman/Doyle/Fuller model with addition of isothermal, adiabatic,
cold, ambient and hot conditions.
Under isothermal and adiabatic conditions, the best geometries are interdigitated
and gear geometries due to higher battery performance and low temperature values
relatively to the other geometries and the main reason for this behavior is the smaller
separator thickness, lower distances for the ions to move (d_max) and higher contact
surface area of the electrodes. For cold condition (265.15 K), the best battery
performance is obtained for the gear geometry and for other conditions (ambient and
hot), the best results are obtained for the gear and interdigitated geometries.
The generated heat is due to the internal resistance related to the maximum
distances that ions move until its intercalation (d_max) and also to the thickness of the
separator.
Thus, it is shown how battery performance can be optimized for specific geometries
taking into account different thermal conditions.
8. Computer simulation of the effect of different thermal conditions
204
8.6 References
1. Dunn, B., H. Kamath, and J.-M. Tarascon, Electrical Energy Storage for the
Grid: A Battery of Choices. Science, 2011. 334(6058): p. 928-935.
2. Lin, X., et al., High temperature electrical energy storage: advances,
challenges, and frontiers. Chemical Society Reviews, 2016. 45(21): p. 5848-
5887.
3. Turner, J.A., A Realizable Renewable Energy Future. Science, 1999. 285(5428):
p. 687-689.
4. Varun, I.K. Bhat, and R. Prakash, LCA of renewable energy for electricity
generation systems—A review. Renewable and Sustainable Energy Reviews,
2009. 13(5): p. 1067-1073.
5. Ibrahim, H., A. Ilinca, and J. Perron, Energy storage systems—Characteristics
and comparisons. Renewable and Sustainable Energy Reviews, 2008. 12(5): p.
1221-1250.
6. Ozawa, K., Lithium Ion Rechargeable Batteries: Materials, Technology, and
New Applications2012: Wiley.
7. Park, J.K., Principles and Applications of Lithium Secondary Batteries2012:
Wiley.
8. Tarascon, J.-M., Key challenges in future Li-battery research. Philosophical
Transactions of the Royal Society A: Mathematical, Physical and Engineering
Sciences, 2010. 368(1923): p. 3227-3241.
9. Tarascon, J.M. and M. Armand, Issues and challenges facing rechargeable
lithium batteries. Nature, 2001. 414(6861): p. 359-367.
10. Scrosati, B. and J. Garche, Lithium batteries: Status, prospects and future.
Journal of Power Sources, 2010. 195(9): p. 2419-2430.
11. Khateeb, S.A., et al., Design and simulation of a lithium-ion battery with a
phase change material thermal management system for an electric scooter.
Journal of Power Sources, 2004. 128(2): p. 292-307.
12. Buchmann, I. and C.E. Inc, Batteries in a Portable World: A Handbook on
Rechargeable Batteries for Non-engineers2001: Cadex Electronics.
13. Balakrishnan, P.G., R. Ramesh, and T. Prem Kumar, Safety mechanisms in
lithium-ion batteries. Journal of Power Sources, 2006. 155(2): p. 401-414.
8. Computer simulation of the effect of different thermal conditions
205
14. Wang, Q., et al., Thermal runaway caused fire and explosion of lithium ion
battery. Journal of Power Sources, 2012. 208: p. 210-224.
15. Zhao, R., et al., A review of thermal performance improving methods of lithium
ion battery: Electrode modification and thermal management system. Journal of
Power Sources, 2015. 299: p. 557-577.
16. Zhao, R., J. Liu, and J. Gu, The effects of electrode thickness on the
electrochemical and thermal characteristics of lithium ion battery. Applied
Energy, 2015. 139: p. 220-229.
17. Park, M., et al., A review of conduction phenomena in Li-ion batteries. Journal
of Power Sources, 2010. 195(24): p. 7904-7929.
18. Du, W., et al., Effect of cycling rate, particle size and transport properties on
lithium-ion cathode performance. International Journal of Heat and Mass
Transfer, 2010. 53(17–18): p. 3552-3561.
19. Miranda, D., et al., Computer simulations of the influence of geometry in the
performance of conventional and unconventional lithium-ion batteries. Applied
Energy, 2016. 165: p. 318-328.
20. Yeow, K., et al. 3D thermal analysis of Li-ion battery cells with various
geometries and cooling conditions using Abaqus. in Proceedings of the
SIMULIA Community Conference. 2012.
21. Cai, L. and R.E. White, Mathematical modeling of a lithium ion battery with
thermal effects in COMSOL Inc. Multiphysics (MP) software. Journal of Power
Sources, 2011. 196(14): p. 5985-5989.
22. Mills, A. and S. Al-Hallaj, Simulation of passive thermal management system
for lithium-ion battery packs. Journal of Power Sources, 2005. 141(2): p. 307-
315.
23. Sabbah, R., et al., Active (air-cooled) vs. passive (phase change material)
thermal management of high power lithium-ion packs: Limitation of temperature
rise and uniformity of temperature distribution. Journal of Power Sources, 2008.
182(2): p. 630-638.
24. Pesaran, A.A., Battery thermal management in EV and HEVs: issues and
solutions. Battery Man, 2001. 43(5): p. 34-49.
8. Computer simulation of the effect of different thermal conditions
206
25. Rao, Z., et al., Experimental investigation on thermal management of electric
vehicle battery with heat pipe. Energy Conversion and Management, 2013. 65:
p. 92-97.
26. Gören, A., et al., High performance screen-printed electrodes prepared by a
green solvent approach for lithium-ion batteries. Journal of Power Sources,
2016. 334: p. 65-77.
27. Srinivasan, V. and J. Newman, Discharge Model for the Lithium Iron-Phosphate
Electrode. Journal of The Electrochemical Society, 2004. 151(10): p. A1517-
A1529.
28. Northrop, P.W.C., et al., Efficient Simulation and Model Reformulation of Two-
Dimensional Electrochemical Thermal Behavior of Lithium-Ion Batteries.
Journal of The Electrochemical Society, 2015. 162(6): p. A940-A951.
29. Miranda, D., et al., Computer simulation evaluation of the geometrical
parameters affecting the performance of two dimensional interdigitated
batteries. Journal of Electroanalytical Chemistry, 2016. 780: p. 1-11.
30. Gerver, R.E. and J.P. Meyers, Three-Dimensional Modeling of Electrochemical
Performance and Heat Generation of Lithium-Ion Batteries in Tabbed Planar
Configurations. Journal of The Electrochemical Society, 2011. 158(7): p. A835-
A843.
31. Bae, S., et al., Quantitative performance analysis of graphite-LiFePO4 battery
working at low temperature. Chemical Engineering Science, 2014. 118: p. 74-
82.
32. Wang, C. and J. Hong, Ionic/Electronic Conducting Characteristics of LiFePO4
Cathode Materials: The Determining Factors for High Rate Performance.
Electrochemical and Solid-State Letters, 2007. 10(3): p. A65-A69.
33. Shetzline, J.A. and S.E. Creager, Quantifying Electronic and Ionic Conductivity
Contributions in Carbon/Polyelectrolyte Composite Thin Films. Journal of The
Electrochemical Society, 2014. 161(14): p. H917-H923.
34. Prada, E., et al., Simplified Electrochemical and Thermal Model of LiFePO4-
Graphite Li-Ion Batteries for Fast Charge Applications. Journal of The
Electrochemical Society, 2012. 159(9): p. A1508-A1519.
8. Computer simulation of the effect of different thermal conditions
207
35. Okamoto, E., et al., Analysis of Heat Generation of Lithium Ion Rechargeable
Batteries Used in Implantable Battery Systems for Driving Undulation Pump
Ventricular Assist Device. Artificial Organs, 2007. 31(7): p. 538-541.
36. Solchenbach, S., et al., A Gold Micro-Reference Electrode for Impedance and
Potential Measurements in Lithium Ion Batteries. Journal of The
Electrochemical Society, 2016. 163(10): p. A2265-A2272.
37. Aurbach, D., et al., Design of electrolyte solutions for Li and Li-ion batteries: a
review. Electrochimica Acta, 2004. 50(2–3): p. 247-254.
38. Zhu, J.G., et al., A new electrochemical impedance spectroscopy model of a
high-power lithium-ion battery. RSC Advances, 2014. 4(57): p. 29988-29998.
9. Conclusions and future work
209
9. Conclusions and future work
This chapter presents the main conclusions of the present work, devoted to the
optimization of lithium-ion battery performance through computer simulation. Further,
some ideas for future works area also presented.
9. Conclusions and future work
211
9.1 Conclusions
Rapid technological advances in portable electronic products (mobile-phones,
computers, e-labels, e-packaging and disposable medical testers, among others) and
hybrid electric vehicles (HEVs) or electric vehicles (EVs) lead to an increasing need for
larger lithium ion battery autonomy with high-power and capacity.
Typically, in order to increase the performance of a battery (power and energy
density), new materials for electrodes (cathode and anode) and separators are developed
and new geometries are explored.
Computer simulations of battery performance are an essential tool for
understanding the main parameters that affect battery behavior before fabrication. Thus,
it is important to develop simulations for optimizing battery performance as these
simulations allow to predict the factors that affect battery performance. In this work, the
effect of the geometrical parameters of battery separator membranes (porosity,
turtuosity, Bruggeman coefficient and thickness) were first simulated. Then, the optimal
percentage of binder, active material and conductive additive in lithium-ion battery
cathodes was evaluated. The choice of battery geometry is important for implementation
into devices and therefore, the influence of the geometry of the battery in their
performance was evaluated at different thermal conditions. Thus, these simulations
allows to develop lithium ion battery prototypes with higher performance for different
applications. It is important to refer that the simulations should be developed according
to the final application of the battery as well as according to its fabrication process.
In this work it has been demonstrated that the ionic conductivity of the separator
depends on the value of the Bruggeman coefficient, which is related with the degree of
porosity and tortuosity of the separator membrane. The optimal value of the degree of
porosity should be above 50% and the separator thickness should range between 1 μm at
32 μm for improved battery performance.
The optimization of the electrode formulation is independent of the active material
type and the minimum value of n, defined as the percentage of binder
content/percentage of conductive material, is 4 at 1C discharge rate, the minimum value
of n depending on the discharge rate. Also, the electrical conductivity of the cathode
depends on n and on the electrical conductivity of the conductive material, being
therefore relevant the selection of the conductive material.
9. Conclusions and future work
212
The influence of different battery geometries (conventional, interdigitated,
horseshoe, spiral, ring, antenna and gear) was studied in order to tune battery geometry
for specific applications.
Maintaining constant the area of the different components, the interdigitated
geometry reach the higher delivery capacity at medium and high discharges rates. The
delivered capacity depending on geometrical parameters such as the maximum distance
of the ions to move to the current collector, distance between of current collectors and
the thickness of the separator and electrodes.
The effect of the geometrical parameters of interdigitated batteries, including
number, thickness and length of the digits, was evaluated and the delivered capacity of
the battery increases with increasing the number of digits as well as with increasing
thickness and length of the digits.
With respect to the different thermal conditions (isothermal, adiabatic, cold, regular
and hot conditions), the gear and interdigitated geometries shows the highest delivery
capacity at medium and higher discharge rates.
In conclusion, the theoretical simulation presented in this work allows to
understand and optimize the components of the batteries before experimental
implementation.
9. Conclusions and future work
213
9.2 Future work
The theoretical simulation of lithium-ion batteries represents a strong growing
research field with the objective of optimizing their performance before experimental
implementation. Once conventional and interdigitated geometries are strongly
implemented in the manufacture of lithium ion batteries and following the thermal
study, it is important to evaluate the influence of the thermal conditions on the
performance of these two geometries when different cathode active materials are used
(LiFePO4, LiMn2O4 and LiCoO2). Thus, at different thermal conditions it can be find
the most suitable active material for the cathode in both geometries (conventional and
interdigitated) in order to obtain high battery performance.
Following the present work on the effect of different thermal conditions of the
performance of lithium ion batteries, it will be relevant to evaluate the battery
performance at different external conditions, such as pressure.
In order to further improve the theoretical models applied to lithium ion batteries, it
will be necessary to develop simulation studies at different scales in order to better
understand the physical, chemical and electrochemical processes and phenomena
associated with the operation of the batteries. As example, the process of
insertion/extraction of lithium ions and the overall battery operation and ionic
diffusion/conductivity through of separator membrane can be studied from different
points of view and at different physical and chemical scales: nanoscale, mesoscale,
microscale and macroscale. For all models at the different physical-chemical scales,
there are a number of relevant variables particularly relevant for battery performance.
It can be also explored the development of specific designs of batteries for areas
such as energy harvesting.
With the scarce lithium resources and the emergence of sodium and magnesium
batteries, it is important to develop theoretical models for these new batteries. Thus, it
would be interesting to applied the methodologies developed in this work for sodium
and magnesium ion batteries. Thus, it would be possible to understand the similarities
and differences between the various types of batteries (Li-ion, Na-ion and Mg-ion) in
order to be used in different application areas.